Is there a food that is halfway between a cheesecake and regular cake?
Perhaps you're looking for the soufflé cheesecake, which has a moderate amount of flour in it. This style is also popular in Japan.

The other style popular in Japan is the "rare cheesecake", which is set with gelatin instead of being baked. I think this is probably denser than what you're referring to, so I left it out of my first edit, but worth considering if the soufflé style doesn't match your expectations.

In Germany, some quark cheesecakes have flour in them as well, and those often have a more sturdy texture with a bit more air than the New York style.