It is not a minor detail, Verizon and Sprint were two absolutely massive carriers and it was truly impossible to run the iPhone on those networks. And running it on T-Mobile meant voiding your AT&T contract and paying some big fees if buying it first-hand. Practically speaking, to most consumers it was only AT&T.

It is not being pedantic to point out someone is saying something massively untrue.

Your comment makes it sound like anyone could just easily take an OG iPhone and run it on any network, but in reality, to US consumers it was only a few smaller networks and only after breaking an expensive contract. See how that's more than just a pedantic difference?