Necessity to define co-NP in the first place?

Question

I've recently started to deal with complexity theory and I'm trying to wrap my head around all the definitions and why they make sense.

One thing I don't quite understand is the importance/necessity of co-NP as its own set/class of problems. Couldn't for example TAUT be classified as a NP-complete problem just like SAT or is there some specific motivation to not put complements of a problem in the same set as the problem itself?

I'm sorry if this has been asked before or if my question is silly and doesn't make sense at all (In this case I'd like to know why if possible), I've really tried to look for it beforehand on the internet.

Kind regards,