Why do you read the entire rule, and not simply the first clause? That's why.
I know it seems simple. I know it's something that hammer into the heads of 1Ls (and thereafter). But it's still something that doesn't always take.
Here, it results in a dismissal of a notice of appeal (and, as an aside, the cross-appeal). Because FRAP 4(a)(4) does give you more time to file your appeal if the other side files a motion for attorney's fees. But, as the last clause of the relevant subparagraph reveals, that's only "if the district court extends the time to appeal under Rule 58." Which didn't happen here. Hence, waiting until the fee motion was decided (six months later) and thereafter filing your notice of appeal isn't good enough. Appeal dismissed.
From the facts of (and various filings in) the case, you can perhaps see why appellant's counsel got confused. Still, you gotta file your notice of appeal at the right time. Get some help if you need it. 'Cause it's important (as in, jurisdictionally important) to get it right.