Part of this question already exists on Reddit, but I thought it would be worth asking again here because it might produce some more reliable and complete answers. In Wiktionary's declension table (here) for unser it lists three variants for the accusative masculine, unseren, unsern and unsren. There are similar variants listed for every entry which has an ending, and the declension table for euer shows the same pattern. I gather the reason is that the ere combination is often slurred in spoken German, and if this slurred version is recorded then it is written as either er or re, is this correct? Is the choice of unsern vs. unsren a regional thing, does it depend on the situation, or are they different spellings of the same word? I'm assuming that unseren is never wrong, though it may be a bit formal in some situations, is this correct or are there circumstances where you must use either unsern or unsren? Is it fair to call these variants contractions or are they considered colloquialisms? DWDS also lists these variants in examples, but it doesn't really explain the differences either, at least not that I could find.
As correctly assumed, unseren is the complete form and the others are shortened. The described shortening is called Elision, and it has several causes:
- achieving a rhyme or metre/meter
- Sprachökonomie, the tendency to remove stuff considered as non-essential, for convenience reasons, which may cause a change in the language in long term.
Unsren has a more poetic/dated touch and I would not expect to find it in new texts, while unsern will be encountered frequently in colloquial context.
Note, that comparatives are especially susceptible as seen in größeren, kleineren, schöneren. I observe from 19th century books, that größern, geringern etc. was considered writable and printable at that time (small example here).
I wouldn't say that there is a difference between them. I think it's just personal preference, maybe some little regional influences as well. You are right, unseren is never wrong and formal always correct. The others are a bit more colloquial but not that much that you absolutely couldn't use them formally. I'd still recommend to stick with unseren in written language.