OK .... firstly, you have to understand that you do not own the cat. THE CAT OWNS YOU!
My wife and I, and soon will be our new daughter too, co-habitate with 2 cats - a long-haired Tuxedo, and a purebred Himalayan. The Tux was born outdoors, and has a really wild streak in her. She is very affectionate, but if not let out daily to kill stuff and visit her friends, and even taunt a few of the neighborhood dogs, she gets very hyper and destructive .... she needs to get out and spend that energy! She never strays very far from sight, and usually wants back in within a couple hours - many times with a mole or field mouse gift for us. (When cats bring home their kills, it is the highest form of affection they can show you!!)
Our Himalayan on the other hand, is a total diva priss, and she knows it! She has once or twice dared to run out when a door is opened, but once outside she FREEZES and panics, as she is not sure what to do next?!? Himmies by nature are very particular. She is very content to sleep all day on her perch atop of the 6' cat condo we have for them, and they say also that Himmies are not fighters like other breads of cats are. Even though she has her claws, I don't think she knows how to use them.
TO answer your original question, watch your cat's mannerisms and body language. If he/she is a feline that needs to run and kill things, you will see it as they will have all kinds of nervous energy, and will make noises by the door like they want to go out.
Other cats could not care less, and will be very content to spend all day and night inside with you.
But for the love of god ..... make sure to "fix" your cat before you let them out to play!!