Well, if everyone actually is having fun, it might not matter- but it certainly matters if a couple is meeting other partners for sex and different participants have different expectations about what's about to go down.
Like if it's a hotwife couple meeting someone new and that guy starts trying to dominate and humiliate the husband that would probably lead to awkwardness at the very least, especially if the new guy is specifically/especially turned on by that kind of play (in which case he's likely to walk away dissatisfied).
It's not just a name; I think there are distinctly different dynamics between cuckolding and hotwifing, although it looks like my own definitions vary a little from some folks here.
The way Alex and I look at it, we're strictly cuckold because she is the dominant partner, she humiliates me privately and publicly, she compares my dick to the much bigger dicks of her partners, and she has a long-term feminization project going on where she's slowly making me her sissy slut. I don't think you're gonna meet a lot of "hotwife" couples who do that stuff.
I think where we differ from some other folks here is that even though Alex is the dominant partner, we both have an equal say in terms of whom we meet and what our boundaries are. If either of us is uncomfortable with someone or something going on, we raise the flag and fix it. Our wife sharing play is for both of us, and if one of us isn't having fun, then we don't see the point.