Is there a characterization of graphs without a K6 minor? (or no K6, K3,4 minor?) Maybe similar to planarity?

Question

It is known that planar graphs are those graphs having no K5 and K3,3 minor. Similarly, outerplanar graphs are those that have no K4 and K2,3 minors.

However, what about those graphs that have no K6 minor? Or no K3,4 minor, or neither of them?

I have found here that maybe there is no characterization so far for having no K6 minor?

Thanks!