My 81 Honda Prelude’s recently started doing this, and I don’t know exactly what could be the cause. When I go to start it, it’ll act like the battery’s low, but it’ll still turn over quickly. Sounds simple enough, but here’s where the bizarre part comes in. If I open the door with the engine off, the dome light glows dimly, again, normal if the battery’s going bad, however, if I flip the switch to turn the light on, it glows brightly, as if there’s nothing wrong with the battery. Opening the trunk, I get the same dim glow as I do when I open the door. Could the problem be a single bad connection of some kind (and are all these items on the same circuit)? Or am I looking at multiple problems?
I’m no expert, but it sounds like a ground problem. You might start by cleaning the batt. connections and the ground wire from the batt.
First a question. How does a battery act low, and yet turn over quickly? Usually the first indication of a low battery is a slow crank. 
Why do you say it acts low?
Now on to your dome light issue. This one is probably very easy. Open the car door, find the pushbutton for the interior lights (usually on the forward jam) remove the screw. Either the screw is loose, or the hole is rusted (get a bigger screw) or the switch is corroded (clean, replace or swap with a different door)
The light has power sent to it, and gets it ground when you open the door, when you flip the switch the ground is at the dome light.
Like A.R. Cane it is a ground issue.
That’s what I get for posting when I’m tired. It’s a slow crank, but the engine catches and fires quickly. So it takes maybe a half second longer to start the car than it normally does. The battery light also winks out with all the other indicator lights when the engine starts up, so the alternator isn’t having to charge the battery any more than it normally would for a start.
Would the trunk light be on the same circuit?
I understand completly. I think you are facing three problems. A low battery, and two bad grounds.
Do you have a volt ohm meter?
For the power feed yes, for the grounds I doubt it. Usually the way a dome light circuit works is power is feed to the bulb, and hot wires go from the bulb to a switch at each door and at the trunk. The switches are open circuit when the door is closed, and when the door is open the switches close and complete the circuit. In I think every car I have looked at the uses door switches, they connect to ground through the mounting screw.
What confuses most people trying to diagnois this circuit is that when the switch is in the door closed position, there is voltage on the wire to the door. When the switch is in the door open position, the bulb becomes the load in the circuit and uses up all the voltage in the circuit, so there is no voltage on the wire to the door.
They don’t understand why NO voltage = light on, and voltage = light off. It is because a load does not become a load until current flows.
Low and high beam headlight problems can also be a lot of fun for the peanut gallery when the trouble shooter has never learned how they are wired on many ( most ) cars. The question is how long you let them tilt at the windmill. Bawahahaha
Well, I figured out what the problem was, without even having to dig out my meter. Turns out that the previous owner replaced the fusible link with some wire. Said wire apparently wasn’t up to snuff and slowly burned away. Now, I just have to figure out if my local AutoZone has the replacement part.