Lights coming on is a bad indicator of how much charge a battery has. Lights do not pull much at all.
The old battery probably has a dead cell. When that happens you won't be able to start the vehicle with just a jump.
If the vehicle starts only sometimes with the new battery in it then you need to do a few things.
#1. Clean the cable connectors very good with a wire brush and confirm the cables are not frayed at the cable clamps. (this fixes the problem most of the time)
#2 Check the ground cable to the engine block. (be sure it is connected tightly)
#3 Check the hot cable to the solenoid to be sure it is not lose.
If all those check out ok then the solenoid on the starter may be faulty. The clicking you heard before is actually the solenoid. It'll do that when it is not getting enough power. The solenoids job is to engage the bendix with the flywheel prior to it also energizing the starter motor.
I'm betting the cables are just dirty.
When I was in college I worked as an associate at Batteries Plus (though far from a mechanic). So I've dealt with hundreds of can battery installs over the years. Without physically seeing the vehicle , I'm going to actually go out and lean towards this being a starter related issue. Especially with the intermittenent starting. It's not taking a jump etc and after it sat for a bit it started etc.
I don't think its terminal/cable corrosion related. You REALLY have to have a lot of corrosion on the cables to really affect a good batter from starting, especially on top terminal batteries, side terminals are a different story. I've see a green christmas tree on top of a '98 Dodge Neon battery and it was still able to start (though just barely). And if there is a lot of corrosion, its generally a sign of a bad battery. The batteries will begin to get really hot during use (car running/driving/alternator charging it) and the acid will begin to boil and either 1) vent large amounts of hydrogen gas through the side vents and/or 2) boil enough to spill out the top caps a bit. Once the acid starts getting to the led terminals and cables, it becomes christmas time.
The Villager's V6 doesn't take a whole lot of power to turn over the engine. It takes a Group 35 battery rated about 450 CCA to do the job, though to turn it over in a normal climate takes a bit less CCA. Interstart/Exide/East Penn batteries in Group 35's range from 500 to 550 CCA, more than enough to do the job. About 95% of your aftermarket batteries are made by these 3 companies, no matter what the label on the side says.
If it was alternator related, either the voltage regulator or the whole unit in general, we would see different symptoms I believe. A bad voltage regulator would definantly harm the battery, but it should still take a jump start. A bad alternator would take a jump, but the car would more than likely shut off after the engine is running since its drawing electricity away from the battery and once the battery cannot fulfill the electrical need of the car, it will shut off everything. Once a vehicle is start, the battery should just sit back and take a charge, unless the alternator (or a cable short somewhere) is draining the battery.
It's possible there is a short in the cars wiring as well. If there is, the amp requirement to turn over the engine may be astronomical. I've seen vehicles with faulty fog light wiring that have maxed out a 100 amp charger trying to do a jump start and still failing to start it.