Avistar, Smarkpark and Longterm (not the PA on site lot), provide valet parking--they park your car upon arrival and shuttle pax to the various terminals; on return, you call them and they shuttle you back. So, you must leave the
key with them. I haven't used Avistar in a while, and never used LT, but Smartpark, photographs your car--this (presumably) will document any damage they caused.
With the "real" Longterm lot (owned by PA), you lock and park your car (and hence keep the key)--however, even though they have patrols, anyone would have access to your car once they are on the LT parking lot property. [Of course, if you leave your car home and take taxis, your car would be safe from damage at JFK.

I've priced the lots you've mentioned. It looks like the cheapest is the Smartpark. I don't want to drag baggage around trains and everything so I will try this one. Unless someone has had a bad experience with them?