Yes, it's context-dependent for me.
I've voluntarily "promised" to muck out the cat's litter tray today. I'm fairly confident I'll manage that. Didn't actually say "I promise," but I strongly implied it as a commitment, which is much the same thing. It's a very small commitment of course. I rarely promise anything, as I don't like the risk of letting anybody down, and I rarely know for sure how anything will turn out. So I usually just say that I'll try. But there can be a nobility in it, giving your word and keeping it. It would be a shame to go through life and never show a trace of that.
When people promise me things, I often think "we'll see about that." But it depends on the person and the promise. To a degree, I'm impressed when they keep the promise. I think it's good for strengthening bonds. It's good when somebody says they're going to do a thing and then they do it.
I usually hate it when people try to extract promises, but if somebody's close very to me then I might accept the demand, depending on what it is. Of course it has to work the other way round too.
I think it's important not to be too literal about promises. Circumstances can change and reduce the need to keep it.