Add a context to each weight

I need weights to sort nodes when displayed within taxonomy (where sort by dates is not suitable, as content is organized by content importance, not by publishing dates). Some of these nodes are published also on my front page (among other news / posts, so visitors see immediately what to pay attention to, as otherwise those nodes could easily be missed within all of the content...).

I came to the situation now, that one of such nodes was positioned relatively hight (weight -77), but I need to add a node belonging to some other taxonomy now, which should have much lower weight (to fit into another taxonomy ordering). The perfect situation would be to have two weights for each node, - one for a taxonomy system (or, even per taxonomy term) and another one for the front page.

For my particular case it would (at least for now) be enough also if weights would simply be neglected at the front page (making all posts to be ordered by publishing date only), but it doesn't seem to be an option. Or, am I missing something?

If it is not possible, what should I change in the code just to skip ordering by weight for the front page (for a quick solution I need ASAP - probably not much to change, but I have no idea where to do that and what exactly to put as a condition - and, I really wouldn't like to break something)?

On the long term, I would consider a context for weight - so we can add arbitrary weights in pair with a context, which could be "global" or more limited - "for a taxonomy system", "for certain taxonomy term only", "for a front page only" etc. (so one node could have different weight when displayed in one "location" and another weight when displayed somewhere else...). Am I asking too much?