I don't know who first coined the name of the shoes as the red shoe theory has been around for a while so I'll just put the story where I first found the term
The Game That is Parenting (Shouta Gets Ten Points) by ShadeCrawler on Ao3
Waking the next morning Izuku had gotten ready for the day before checking out and headed to a small twenty-four-hour American-style diner, he connected to the cafe's wifi and started a video call with his mom, he had to wait for a few moments before he and mom connected.
He beamed upon seeing his mom "Good morning mom!" He cheerfully greeted "Hello Izuku! How are you doing? Did you leave the hotel already?" Inko responded likely noticing the booth he was sitting in instead of the nice little room he had booked. "Yeah, I'm about to have breakfast then I'll be heading to Kumamoto." Izuku answered before briefly telling a waiter what he wanted.
"How are you doing? Did you sleep alright last night?" He inquired turning his attention back to his mom "Yes I did. Did you honey? I know you have trouble adjusting." she questioned settling down at the kitchen to eat her own breakfast "Yup, had kind of a weird dream about Wash taking me to see Atlantis in one of his soap bubbles. I think I was pretty worn out from yesterday's excitement." Izuku replied rubbing the back of his neck.
The two happily went on to talk about the various mermaid versions of heroes he saw in his dream as they ate breakfast together, Inko told him she was thinking of taking some time off and maybe going out on a girl's day trip with Auntie Mitsuchan, Izuku tried to encourage her to do it and even go to a spa since he couldn't remember the last time his mom took some time to take care of herself.
Inko gave a quick goodbye after about half an hour as she had to get to her job and Izuku promised to call again if he could during her lunch break, Izuku finished up his breakfast then left to get to the next prefecture, Kumamoto was quite a bit further and he wanted to be there with enough time to relax for an hour at least before throwing himself into the districts.
He took a break about three hours in to again stretch his legs and have a bathroom break but before he got back on the road, he texted with Uncle Masachan and got him to help convince Auntie to take his mom out for some fun, Izuku was tempted to ask how Kacchan was doing but ultimately decided against it since it was a can of worms he wasn't sure he could deal with it and do his work.
Izuku greatly enjoyed crossing the Kanmon bridge and the view was simply amazing then he made a mental note to find some time to come back to take some pictures here, he decided against going to the hotel he booked right away this time and looked for an internet cafe, Izuku found a nice little place by the Shirakawa river and ordered something to snack on while he started his college courses.
When it was his mom's lunch break he called her up and talked with her as he did his work, he took a moment to show her the river as well and promised to grab a souvenir for her, when he was done with his classes and he had a very satisfying info binge on how quirks had evolved from environments people were born in as well as parental factors.
He paid for his time and the food he ate before moving on to visit the districts like he had planned, given the people in Hiroshima were only vaguely aware of him he didn't expect much of a welcome and Izuku was mostly right, a good majority had no clue about him or his work and he was definitely considered an outsider.
Izuku was looking around the last red-light district for a place to teach when he ran into a member of the local chapter of the Russian Vipers, the brunette was quick to snap at him "'Ey! Watch where the fuck you're going shithea-" but he suddenly cut himself off when he looked down to find the pack of cigarettes he dropped, he blinked and his eyes got wide "You a two-toes?" He asked suddenly taking Izuku off guard.
Izuku blinked also shocked, he had heard the term thrown around a lot online in the one or two quirkless support groups he was in way back before his mom found out, he hadn't really participated in the groups as he was far too focused on taking care of himself after what happened at school, he did try to message there once and a while originally but he just lost interest trying to find some time to keep up with the group years ago.
But he had never met anyone that knew about the name two-toes outside of those groups though so he was a bit less hesitant to answer him "Y-yes?" Unsure whether or not he should be scared of the man turning on him, however, the man surprised him by suddenly laughing and apologizing "Fuck, sorry I didn't mean to be a bitch. Especially not to a two-toes, got a little two-toes myself you know."
Izuku almost gasped "R-really!?" He exclaimed "Yeah, fuck it was a pain in the ass to find the kid shoes. I'd know those damn shoes anywhere." The man joked, running a hand through his hair "C-could I meet them?" Izuku requested, he hadn't ever met anyone quirkless close to his age, the last time he could ever remember meeting anyone quirkless in person was one time when he was really young, he had met an elderly lady on a train with his mom.
Well, Izuku didn't think he had but it was hard to tell quirkless people from quirked outside of the bright red primordial shoes and not everyone had shoes in red-light districts... or wanted to be known as quirkless, he'd to ask some of his students about that the next time he was in the South Side.
"Sure, probably be good for the squirt to meet another two-toes his age." The man responded as he picked up the box of cigarettes and Izuku was shocked that the man agreed so easily "R-really!? J-just like that? You don't even know m-me!" Izuku replied out of disbelief and the man scoffed.
"You haven't been around many two-toes have you?" He asked and Izuku nodded, there were offers to meet up in person from the group chats, but he didn't go to any, he didn't want his mom to waste the time or money to bring him to one "Well if you did, you'd know that two-toes like you stick together like a pack of rabid dogs. So even if you're a devious little bag of shit, I know you won't be a fuckhead to the kid." The man told him to brush the whole thing off.
Izuku wanted to question how many quirkless people the man had met but pushed it aside as he quickly followed the man to an apartment building across from a vandalized park "Stay here, I'll get the little bastard." The man said and left Izuku standing in front of the building, he checked the time and he still had like an hour before he needed to start teaching so he'd be able to talk to the kid for a few minutes.
Looking back at the park he could see there was a good-sized clearing by a veranda, the veranda was also in good shape, but he'd need to check it for any gang claims or drugs inside of it, upon hearing the door to the building open he turned and saw the brunette man had returned with a little boy about ten following behind him.
Izuku's eyes went straight to the boy's shoes and saw that while they weren't as vivid as his own he was wearing primordial shoes, nervous excitement was clear in the other boy but he was holding back from running up to him "H-hello!" Izuku warmly greeted waving to him, the boy practically lit up when he did but didn't move any closer.
The man scoffed giving the kid a soft push "Well go on, you've met other two-toes before and he ain't gonna bite your damn head off." He gruffly encouraged bemused as he watched the boy run over to him "H-hi! I-I'm Osamu!" The boy told him and Izuku was reminded of the kids back at the South Side "I-it's nice to meet you Osamu-San. I'm Midoriya but you can j-just call me Deku-Sensei." Izuku introduced himself much to the boy's confusion.
"But a-aren't you too y-young to be a teacher?" Osamu questioned head tilted and Izuku rubbed the back of his head "Y-yeah, but I'm t-tutoring a lot of p-people so that's become my nickname." He explained choosing to leave out the reason he came up with it was so gangs wouldn't go around spreading his name in the underground.
"You must be r-really smart then D-Deku-Sensei." Osamu replied making Izuku flustered "W-well I guess... I mean I think I-I'm doing alright in college... " he muttered and Osamu gasped "You're in c-college!" The boy exclaimed in absolute awe of the fact, Izuku couldn't help being embarrassed as the boy seemed to have stars in his eyes as he looked up at Izuku.
"U-um, yeah? I-I'm working on getting a q-quirk theory degree but I'm in an o-online college, they don't h-have an age requirement f-for it." He told him but the awe hadn't left the boy "C-can, you go t-to medical s-school online?" Osamu asked and Izuku blinked.
"Um, I... Don't k-know, you can probably do s-some of it on-nline but medicine usually involves a l-lot of direct interaction with people. And doctors and nurses c-can't learn about how to do surgery o-or insert a needle properly w-without practicing it for real." Izuku tried to explain the best he could since he didn't actually know about any of that.
Osamu became a little sadder "Y-yeah, that makes s-sense... " he muttered eyes falling to his feet "Do y-you want to g-get into medicine Osamu-San?" He inquired trying to meet the boy's eyes, the boy became bashful and just nodded, Izuku suddenly understood the boy's shyness of the topic.
Most medical schools in Japan didn't take quirkless students whether they were open about the rule or not, if Osamu really wanted to go into medicine it would be quite difficult but Izuku did have one idea.
"Have you considered b-being an E.M.T? You k-know the people in the a-ambulance?" He inquired and the boy shook his head "I think y-you can become an E.M.T. i-if you finish E.M.T. c-classes and get certified." He suggested, becoming a paramedic was one of the things his mom had suggested before if his dreams of being a hero fell through.
Osamu's eyes got wide "R-really?" He whispered hope clear in his eyes "Y-yeah, I'm pretty sure." Izuku confirmed nodding his head, his mom had double-checked it when he started college, Osamu began grinning and launched himself into hugging Izuku tightly, it was only through experience from the young South Siders that kept the flailing greenette on his feet as the boy began loudly thanking him.
Osamu's father meanwhile happily watched Izuku's suffering from his spot ten feet away as he lit up a cigarette with a very self-satisfied smirk on his lips.