Running is a full round action.
Charging is a full round action.
Withdrawing is a full round action.
Does it make sense to not be able to run and jump in mid-run?
What does the phrase "running start" mean in the Jump skill? What does the phrase "maximum movement" mean in the Jump skill?
All Jump DCs given here assume that you get a running start, which requires that you move at least 20 feet in a straight line before attempting the jump. If you do not get a running start, the DC for the jump is doubled.
Distance moved by jumping is counted against your normal maximum movement in a round.
The issue here is the phrase:
A Jump check is included in your movement, so it is part of a move action.
Can a Jump check be part of a move action? Yes.
Can it be part of some other type of action? Probably yes. This phrase states that it is part of a move action, but does not limit it to only move actions. Otherwise, it cannot be done while running.
So, can a Jump be made while running, charging, or withdrawing? Yes, no, and yes. It is not the Jump skill that prohibits it for charging, it is the charging action:
You must move at least 10 feet (2 squares) and may move up to double your speed directly toward the designated opponent. You must have a clear path toward the opponent, and nothing can hinder your movement (such as difficult terrain or obstacles). Here’s what it means to have a clear path. First, you must move to the closest space from which you can attack the opponent. (If this space is occupied or otherwise blocked, you can’t charge.) Second, if any line from your starting space to the ending space passes through a square that blocks movement, slows movement, or contains a creature (even an ally), you can’t charge. (Helpless creatures don’t stop a charge.)
Withdraw does not have this type of limitation. You can move wherever and however you want up to a double move. You cannot Tumble while Withdrawing because Withdraw is not normal movement, it is a specialized form of movement which prevents an AoO:
Tumble at one-half speed as part of normal movement, provoking no attacks of opportunity while doing so.
The Jump skill does not restrict the movement, nor does it limit you to only doing it for a Move Action. One can interpret the "it is part of a move action" to limiting it only to a move action, but that phrase does not really do that. In fact, if you literally read that phrase to mean that, you could not Jump in the middle of a Double Move (two move actions, not one) either (which is just as silly as not allowing it for a Run action).
My vote: You can Jump as part of Running and as part of Withdrawing, but not as part of Charging.