Ask 10 biologists to define species and you'll get 12 answers, but while a species must be able to have fertile offspring it's always linked to other traits like reproductive isolation from other groups and even more traits after that but it's all disputed and too complicated for a forum comment.

So in the example of hominin interbreeding they'd have been separated by geography prior to the migrations and be different species as a result of that and other factors.

For example, given another ten thousand years of reproductive isolation Old and New World homo sapiens might have differentiated, or maybe not. It all depends on population sizes and whether there's enough adaptative pressure (aka mass dying)

There's a movement now to try and define species based on genetic similarity but I think that's pretty fucking silly when a single mutation could, for example, make the species sterile. You're just never going to able to define it in a mathematically useful way.