First entry to Italy on a Schengen visa issued by Greece [duplicate]

It is to some extent up to the border guards. In principle, it's not a problem except if the change of plans is so drastic as to make them suspect visa fraud.

For example, transiting through Italy on a ticket to Greece is perfectly OK, adding a side trip or deciding to stay a little longer in Italy than in Greece should be fine. On the other hand, going to a conference in Italy on a single-entry visa with no intention of visiting Greece at all (i.e. no ticket, nothing booked there and nothing to support your intent to go there) would strongly suggest you just applied to the Greek authorities to skirt the rules and prevent Italy from evaluating your application.

Those rules are detailed in the Practical Handbook for Border Guards (Schengen Handbook) and in Handbook for the processing of visa applications and the modification of issued visas but you won't find anything in there that would suggest you have an unconditional right to enter.

There are guidelines for determining which Schengen country you should apply to for your visa, depending on your travel plans. If the guidelines suggest you should not have applied to Greece, I doubt anyone would take issue with it. If they do, you can always say that your plans changed after you made the application.

What do you mean by "take a print for records"?