Hi aristarkosis! And welcome to the forum!
I think you have raised a fair queston, so I'll have a stab at an answer.
Humanism is a very wide description, and some don't like the label much more than you do. (Are they the a-labellists...?) For myself, humanism is just the belief that there is no supreme being, but that life, the universe and everything, should be looked at in a positive way, as opposed to a nihilistic way. In other words, we are here, the world and the universe are here; how do we make the best of it?
That is sufficiently wide to accomodate almost every humanist, I'd say. (Or maybe too wide to have any meaning... ) But while I, as a self-described humanist, think we should, for example, care in some way about animals, I can only do that as a human, with human eyes. I'm inclined to think that humans are pretty special, but that doesn't mean I care not a fig for anything else. For much of the time, my perspective is about human affairs though, especially as it contrasts with other, religious, perspectives, because I am a human.
I'd also query why you are concerned about using "humanism" as a term, because you think it may be too restrictive a label, while you seem happy to describe yourself as a "deistic agnostic" and a "unitarian universalist".... Labels are only ever a shorthand anyway, so it is perfectly possible to adopt different ones, as and when the need arises.
Does that help at all?