Not an easy question to answer Dolicus.
No, it isn't the pot that determines a Bonsai. The pot does two things. It contains the tree and also compliments it. The focus should never be on the pot. It should be on the tree.
There are a number of recognized Bonsai styles but the five basic ones are: Formal Upright, Informal Upright, Leaning/Slanting, Semi Cascade and Cascade (as in the sketch below)
Within those categories there are quite a few more, like Root over Rock, Root on Rock, Broom, Windswept, Twin Trunk, Group (or Forest), Raft (or Sinew), Clump, Literati and some other minor styles.
All the styles should conform to the rules governing them. Most aren't that easy to create so some leeway has to be given - trees do what trees want to do. We can only help them on their way.
If your trees are pruned into shapes of animals or geometrical shapes then it doesn't matter whether they're in a Bonsai pot, normal Garden pot or a bucket, they would still be a form of Topiary. A friend of mine grows flowers in Bonsai pots but they couldn't really be called Bonsai.
The Chinese way of styling trees is called Penjing and differs to Japanese Bonsai in that some pretty radical shapes can be produced.
Can you post some photos of your trees? It'd be interesting to see what you've achieved (and I mean that in a nice way). It would give us more of an idea of what we are talking about.
There isn't really such a thing as a 'contemporary' Bonsai as there is in art, but the main thing is that you are happy with what you're doing with your trees.