All the circuit diagrams I try provide the opposite effect I'm looking for. Basically, I want to shut off the circuit when the lid is closed and activate it when the lid is open. Does it have anything to with a reed switch?
You are correct.
A magnetic reed switch is activated whenever it is in close proximity to a magnet.
There are some reed switches with two poles ( one closes and the other opens in the presence of a magnetic field). DPST
The other kind (normally open) are open but close in the presence of the magnetic field. SPST
The type you need is a normally closed magnetic reed switch that opens when the magnet is near. Or buy a DPST and use the normally closed contacts.
You can find magnetic reed switches at Radio Shack just explain what you need to the salesman.
Be sure the current is rated to handle the load or the switch will fail after a few operations.
Hope this helps,