I use Eve Energy smart plugs, which seems to be supported in Home Assistant through the matter integration. Local first, no bullshit remote account requirements, good quality, around 40€ / USD 45.
None of this is cross-compatible.
Correct. Its called split phase, a 240V transformer is center tapped and that tap is grounded to create the neutral. Either end of the transformer to neutral is 120v and end to end is 240V.
> EU 400V (380V) is 3 hots, neutral and ground.
Three phase 230/400: 400V is line (hot) to line with 230V line to neutral. More for industrial use but I hear some homes can have this service for whatever reason.
> None of this is cross-compatible.
Not really. A 230/400 volt system also supplies 230v single phase. A 230 volt European device will work fine on 240V split phase unless it has a motor which will run faster on 60 Hz which could overload it. Though I have a machine with a three phase 380 volt 50 Hz motor running happily on 230/400 60 Hz from an autotransformer supplied by 120/208 60 Hz. Just runs 20% faster.
> Three phase 230/400: 400V is line (hot) to line with 230V line to neutral. More for industrial use but I hear some homes can have this service for whatever reason.
It varies from country to country in Europe. In the UK you'll almost never find 3-phase in a home, in Sweden even apartments usually have 3-phase supply. In my Swedish apartment the only thing connected to more than one phase is the induction hob.