Is the "season" meant to be the interval from September/October to May/June or is it not strictly defined like that?

Bluntly, no, there is no strict definition for a "season".

Now, traditionally, broadcasters would start showing a season in August, and run until May -- mostly because of the belief that people watched less TV in the summer. But this gave them the opportunity to have shows set during Thanksgiving, Christmas, Valentine's Day, etc.

Some shows would even make reference to the "summer break". Castle, for instance, would end the season with a "breakup", only to spend the first episode or two having the characters "make-up". But some shows, especially those ending in a cliff-hanger, might have one or two episodes set "last season" to wrap up the loose ends, before jumping ahead three or four months to "now".

Of course, once all the smaller networks appeared, they saw the huge "rerun" gap left in the summer by the broadcast networks, and quickly stepped in. And the premium networks took a card from BBC and created shorter seasons.