imo

solo staker = anyone who puts up 32 ETH and has full custody of their validator (may even run hardware elsewhere)

home staker = anyone who validates the network with hardware at home. includes DVT stakers and rocket pool operators

staker = anyone who stakes in any way (LST, custodied, whatever)

advocating for independent operators @ethstaker // stake responsibly //
be a kind human // slomad

Would it ever be possible to prove the difference between a solo staker and home staker?

Like building cool things. Collector. Crypto since 2013.

doubtful. any method i can imagine to allow home stakers to be identified or self-identify could be gamed by those not running their nodes at home. but that doesn't mean it's not worth putting thought into!