To cut, as you already know, is generally 'to slice with a sharp object'; if you "cut your own arm", you take a knife and make an incision somewhere on your arm.
To cut out is probably, in this case, 'to free from entrapment by cutting the trapping material'. If you "cut out your own arm", then somehow one arm is trapped, perhaps under a fallen log, and you use your other arm to cut the log until you can get your first arm free. (This usage seems very unlikely.)
The most common use of cut out is 'to shape an item by cutting a larger element'; for instance, if I have a piece of paper, I could 'cut out a paper doll'.
To cut off is to sever or detach; if you "cut off your own arm", then either you no longer have an arm, or you need major surgery to get it re-attached.
"She bit into an apple." This is grammatical and makes sense on its own. In the story of Snow White, biting into a poisoned apple is an important part of the story.
Snow White bit into the fruit, and as she did, fell to the ground in a faint: the effect of the terrible poison left her lifeless instantaneously. (Source)
"She bit off an apple." doesn't make sense without more context. Usually we would say something like "She bit off a piece of apple." To "bite off", a piece needs to be separated from the whole with your teeth. If there was an edible apple tree like the one below, she could bite off an apple (from that tree).
There is an idiom that might help explain it - to bite off more than you can chew. Taken literally, it means that the piece you bit off is so big, it fills your whole mouth making it hard to close your jaw to chew it. I picture something like this
Some more examples:
The dog bit into the man's leg. (Doesn't tell you if the dog has let go, or if the dog has taken a chunk out of the leg.)
The dog bit off the man's nose. (Most or all of the man's nose was separated from his face. Ouch!)
The dog bit the man. (Doesn't tell you where the dog bit the man, or whether it bit anything off, or how severe the bite was.)
Best Answer
jump out of means to leap from a place which is understood to be (or to belong to) an enclosure of some kind.
to jump off means to leap from a place which is understood to be a platform of some kind.
It is possible for the same thing to be understood by one person as an enclosure and by another person as a platform. train is a good example of that. One person might have in mind the steps of the train and say off while another person might have in mind the enclosure of the train car and say out of.
"To jump out the window" is standard but "to jump out the train" is regional. Standard is "jump out of the train". That may have to do with the difference between an aperture and an enclosure. We go out or out of an aperture but out of an enclosure; we don't go out an enclosure.