Originally polysynthesis only meant that an average word had a high content of morphems or meaning elements, without any clear definition how high it should be to call a language polysynthetic.
Later on one tends to call languages that are head marking for at least subject and object as polysynthetic, and one will often add the criterium of noun incorporation. But even if a language has these traits, it does not necessarily have a high content of elements in an average word.
I would define a polysynthetic language as one with both these traits.
Remark that a headmarking language can be dependentmarking at the same time, also called doublemarking, and of cource still be polysynthetic.
Agglutination or fusing of elements have nothing to do wuth the definition, exept that a polysynthetic language must at least be halfly agglutinating to be learnable.
It is very difficult to answar how common this language type is, because many languages have structures that are not so easily classified.
Italian, French and Spanish, for example have a higly templatic verbal group consisting of the main verb, auxiliary verbs, object clitics and adverbial clitics, and these languages are partially headmarking. And these clitics behave more like affixes than just clitics.
So these languages are at least very near to be polysynthetic, and the same holds for modern Greek.
There is also a bias against calling traditional culture languages, like Hungarian, Georgian, Romance languages and Modern Greek polysynthetid, because that would be the same as classifying them togeather with the languages of peoples tagged as primitive.