Good heavens you're all confused.
Jumpseating (with respect to pilots) is a professional courtesy extended by pilots to pilots (with the concurrence of the airline). It allows a pilot who needs/wants to travel to fly on an aircraft and, if there is no seat in the back, to occupy the flight deck jumpseat. It sometimes requires you to fill out a jumpseat form and rarely (ever?) involves the payment of any fee. To occupy the flight deck you must be verified, either through the airlines computers or through CASS, to jumpseat "in the back" you don't need to be verified (depending on the airline).
Non-revving is the act of taking advantage of a program airlines offer to airline employees to fly either for free or on a reduced fee schedule - you are a non-revenue passenger. A non-revenue passenger, like any passenger, cannot occupy the flight deck jumpseat.
If you list as a non-revenue passenger you cannot occupy the jumpseat, however, if qualifed, you can change your status to jumpseater, in which case you cann occupy the jumpseat. A jumpseater can sit up front or in the back, a non-revenue passenger can only sit in the back. It's important to know what you are (and more important that the gate agent knows what you are) - because non-revenue passengers sometimes pay fees to travel that jumpseaters do not - even though they may occupy the exact same seat.
A jumpseater is travelling at the pleasure of the captain and ALWAYS checks in when boarding (I guess there are some airlines that say not to, but stuff that, it's the captain's option, I'll ask for a ride). A non-revenue passenger is riding at the pleasure of the airline and does not need to check-in with the captain and at most places they don't want you to. If you want to check-in have at it, but most crews will look at you a bit funny if you do.
While it sounds like DL has some rules written down somewhere, professional courtesy comes into play with respect to jumpseating when there are multiple pilots competing for the seat. As far as I'm concerned any pilot travelling TO work (regardless of their airline) has priority, then comes any pilots travelling FROM work (again regardless of their airline) and then comes travelling for fun or pleasure. If you're travelling for personal reasons and you take the jumpseat from someone using it to commute, even when you have priority, then you're not a very nice person.