That's what I'd call a PA head. In the good old days bands had one of those with 4 or 6 channels and a pair of speakers and we used them for vocals only. Everything else went out via its backline. Drums were not miced except maybe dropping a mic in the kick drum for a bit of thump. I was using a Phonic brand one until quite recently.

And yes, it's a variant of the powered mixer genus. You can also get desktop types which are often more fully featured.

If you have powered speakers, you don't need a powered mixer. Look for as many mono mic inputs as you can get if you want to use it live. I'd look for a mixer with a swept midrange if you're micing acoustic instruments but it's not essential for vocal use. I've no specific recommendations. There are loads of brands which are all more or less the same. Mackies are good but their cheaper Tapco range could be all you need for a few mics. By and large I'd avoid the smaller Behringers.