I assume you speak about way/205789838. (If not, please inform me)
Usually I don't change the road classification, unless I'm pretty sure about it. In this case it is obviously not a "link" type of highway. By checking the history for this way you can see what changes I made myself and what I left as it was (even if it was probably incorrect). I'm not living anymore on the island so I can't be 100% sure, but I assume the road is open now for traffic.

Is there an official name for these parking lots? I'm asking because on the "traffic signs" (well, more like billboards) around town I saw them listed with different names.

I saw your changes made after my comment, I've also added landuse=residential so now the name should be displayed. Still not happy with the result at Πλατεία Λαϊκής Κυριαρχίας, I guess mapnik renders 'leisure' on top of amenity=parking. I'll think of a solution.

I have doubts that those two public squares should be tagged as leisure=park since, well... there's no park there, neither in front of the theater nor in front of Port Authority. As for an alternative... perhaps highway=pedestrian and area=yes, but for smaller areas than right now. Or even better, a multipolygon.