NP is not an abbreviation for "non-polynomial". You made a very good attempt at explaining P vs NP, but just like most of the other articles that try to make it digestible for a wider audience you screwed up by leaving out the deterministic vs non-deterministic bit.

Sorry for putting it so harsh.

"P problems are in fact contained within the set of NP problems i.e. non-polynomial time solutions exist for P problems. However, all problems in NP may or may not be in P."

Yes, P is in contained in NP. However, the reason is wrong. It is contained in NP because any problem in P is verifiable in polytime (i.e. by actually solving it).