It's an interesting question, we often talk of 'socioeconomic class' but in reality social and economic class are quite distinct things I think. This is particularly pronounced in countries like the UK where the social class system is very entrenched; it correlates with wealth but can be fairly independent of it, it's much more about the subculture you belong to and the way you see yourself within the world. You're born into your social class and you can't really change it within one generation no matter how much wealth you acquire.

Economic class is the more useful framing and it's exactly as you say I think, you either work to put your bread on the table or you own things that put your bread on the table for you. There's other features of economic class but that's the dominant one. This is the class difference that matters the most in my opinion when it comes to explaining aggregate motivations.