The example that SIL (Summer Institute of Linguistics) site gives for polysynthetic is Inuktitut, and the one example from the language they give is characterized by two things:

(i) A large number of morphemes. This is something any agglutinative language can do if you look for extreme examples, e.g. English modal verbs in 'I wasn't going to have been doing it' might be encoded in an agglutinative language as multiple affixes.

(ii) The verb and its object are part of the same word, e.g. 'hunt seals'. This is called incorporating, and English does this too ('babysitter', 'deer-hunting'), but many North American languages have it in a more pervasive way. And those like Inuktitut or Mohawk that are also agglutinative will therefore produce words that contain most of the meaning of the sentence.

I still don't think 'polysynthetic' describes anything very clear, unless it's used as shorthand for incorporating and agglutinative.

Another old term like this is 'holophrastic', where a single word expresses a sentence. You can do this more easily in such languages, but that doesn't mean they don't have distinct words.