But, I can't recall anybody ever giving me any training in what level of understanding of aircraft systems and emergencies I can actually expect of what controllers at what levels?
Never assume that the controller will know more than that which you provide.
State your problem clearly, simply, and succinctly, and state the nature of service that you require.
"
Wonderworld Approach, Cherokee Niner Three Alpha Foxtrot Roundworm."
"Cherokee Niner Three Alpha Foxtrot Roundworm, go ahead."
"
Cherokee Niner Three Alpha Foxtrot Roundworm has an electrical failure, 30 minutes battery remaining, request direct Spongebobville."
"Cherokee Foxtrot Roundworm, Wonderworld Approach clears you direct Spongebobville. Say nature of help requested."
"
Cherokee Foxtrot Roundworm requests early clearance to land, due to limited battery life remaining. Thanks."
"Cherokee Foxtrot Roundworm, roger. Contact Spongebobville Approach now on One Two Three, decimal Four Five Zero."
Done.