put a test light on the starter terminal, the big one and touch the other end to ground. This should be good and bright even with the key out of the car. If its not lit or is dim(compare it to the light across the battery terminals) there is a problem with the cable connections, the cable or the ground. Rarely the ground as the starter is grounded to the transmission/engine by big bolts. It can be the main battery ground cable however or the engine ground(s).
Second, put the test light ground connection on the starter bolt, or other bellhousing bolt, the while touching the test light probe to the small starter bolt, have someone try to crank the car. When and only when they have the key in the start position should this light, but it should light up then.
You can also test the power cable by putting the test light onto the ground and touching it to the big bolt again, and it still should light. But now, you will have to have someone try to crank it again. If it goes noticably dimmer, the cable has too high a resistance due to corrosion, a large pinch etc. I will dim slighty, but if it really dims out, thats your problem.
If you get no light when you touch the big cable, its not connected to the battery.
if you get no light on the small cable while the key is in the cranking position, there is no signal getting from the key to the starter. If the light lights when you have the test light on the small bolt on the starter, and the key is in cranking position, you don't have to worry about the ignition cylinder, the ignition switch, all the wiring from there to the starter, the alarm or anything else.
When I originally posted about replacing the starter, I had assumed you had done these basic tests, and I apologise if that caused you to buy a starter needlessly.
