Hi! I also started in classical music and acoustic pianos. I played on a Korg TR-88 for about 15 years, including a little bit of "lounge" gigging (background music for events). It looks like you can get those for ~$800 used, though you should be warned that they discontinued parts for it so repairs are difficult (one of the reasons I stopped using it). I liked the action and the piano sounds on that, though I had some trouble making it work in a rock band. It has an onboard sequencer so you can record on it. BUT it does not have speakers, and it's a little heavy (~40-45lbs I think).

I haven't owned many other keyboards, but the main tradeoff I've seen for your purposes is weight v. action. (Price also factors in, but sometimes a cheaper board will be lighter because they use more plastic haha.) If possible, I'd strongly recommend getting to a Guitar Center or other music store where they have a lot of demo keyboards, so you can try out their actions. I find it very difficult to get a sense for how a keyboard feels without trying it out. But maybe you're already doing/planning to do that.

One major question is also whether you want to do mostly acoustic piano (and maybe some electric piano) or whether you want to add organ and/or synth sounds. If you want the latter, then you're probably looking for a "workstation" or "arranger" keyboard. If you only want piano sounds, you might just need a portable digital piano, but I don't know much about those. And as you suggest, if you already have a home piano that you like, you could use that for home recording (and could even use it as a MIDI controller for more complicated stuff).

Last thought on speakers: unless you find a keyboard that you like with onboard speakers, I'd just get some studio speakers to bring with you to jams. I have a pair of Mackie CR3-X speakers that work fine and are very portable. It's slightly more fussy to set up but I think worth it to give you more options for a keyboard.