I didn’t say anything about not liking either of these. The two scenarios are qualitatively different. The purpose of the one at home is to learn what happened that day, how the other person feels about it, planning what we do with the rest of our day, and so on. It’s an exchange of information.
The purpose of asking the cashier about their day is not to actually learn what happened with them (unless you actually know the person of course). It is exchanging pleasantries or just making banter, without the intent of exchanging any information that matters to the other person. I don’t dislike it. But it’s not a conversation, it’s small talk.
I read your top level comment as well and you do seem really irked that some people differentiate small talk from conversation. It seems like you’re fighting windmills though, and it’s in fact you who for some reason has strong feelings about the topic.
Small talk is an important part of interpersonal communication, and it’s good when it creates a sense of comfort, belonging, or serves as the prelude for a deeper conversation. But it can be annoying if it’s self serving, because either it fails creating any positive feelings, or it never gets past the warmup phase. I don’t think there’s anything wrong with people who don’t enjoy small talk, or with those who do.