Which type of pump is appropraite for the well

If the pump is a 3-wire submersible with a control box, you can do some basic troubleshooting with an ohm meter without pulling it. If you have a control box like this:

Then you can pull the top off. Inside there are terminals that have black, yellow and red wires that go to the pump. Inside the cover of the control box, it will tell you the resistance ratings (in ohms) between each of the colors, which you can check with any standard multimeter. If those readings are off, then the pump needs to be replaced. If they are OK, the next thing to check is the current draw. You need a multimeter with a current clamp on it, like this:

and you need to be able to clamp around one of the wires while the control box cover is on. The red wire is the starter wire, so it should have a high current then go down. The others should draw whatever the control box says is the normal current draw for the motor. You can measure either the wires going to the pump, or coming from the panel.

If the current draw is very high before it shuts off, the pump is probably jammed. You'll need to pull it and possibly take it apart to fix it.

If there is no or very low draw (and the ohm readings are ok), the control box is probably bad, and you can just replace it without pulling the pump.

I used to work for a company that did that kind of work. Anytime we installed a pump in a cistern like that, we would use a pitless adapter to attach it to the incoming pipe. They're normally used in drilled wells to be able to pull the pump.

If you are lucky, then yours has one too, and all you need to do is screw a threaded pipe into the top of the adapter to be the pull bar (typically 1 1/4"), and then pull it straight up and out. Because you are not using clean potable water, and especially if it hasn't been pulled in a long time, it may require some "gentle" persuasion to get it out. Wiggling and/or using a T-shaped bar (which you can hammer from the bottom) usually does the trick.

Before you put it back you should check the o-ring, and lubricate or replace if necessary. It's not really as important in a cistern as it is in a well, since if it leaks the water just goes back into the cistern, and the pump doesn't have that far to push water anyways, but it's just good practice.

I have run across cisterns where this wasn't the case though. If the pipes come in through the top, you may just be able to pull them and pull the pump out. If not, you may be able to cut the pipe, and then pull the pump from there.

If those methods fail, or if the pipe enters the cistern at the bottom, then the only way to get at it would be as @longneck says and drain the whole thing, then crawl in and get the pump out.

If you do have to drain it, it's a good oppertunity to clean it as well. There is likely a lot of sediment built up on the bottom that you can get out with a shopvac. Spraying it down with a pressure washer probably won't hurt either.

I would highly recommend that before you put the pump back in, you install a pitless adapter.

All sort of issues here - while there's a short list of things that one normally looks for with extreme short cycling (such as a blown bladder in the pressure tank) that doesn't touch the 90 PSI issue...

If, in fact, water is going back down the well, the check valve in the pump (there is pretty much always one built into deep well pumps) is bad. Depending how your system is set up, there may or may not be another one inside the house, which would also have to be bad. My own system, after considerable reading and opinion-gathering, has only the check valve on the pump 300 feet down my well. When it goes, it's pump-pulling time. That would be the case even if there were others, and others might actually cause that to happen sooner according to a number of experienced well folks, which is why the other one I bought is still sitting in a box, not installed. In another few years I'll put a new pump on the shelf for when this one dies, so it's not a panic buy when (not if) I need it.

80-90 PSI is VERY high and almost certainly indicates a problem with the pressure switch. Certainly on my well the emergency relief valve lets go at 100PSI. On 40 off 60 is usually the upper range of domestic well pump "common settings" (30/50 and 20/40 being the other two commonly seen.) Ideally you'd know what your setting usually is or it may be recorded on a sticker somewhere on the system. You mention it going to the very high pressure when first switched on after being off for a while - what was it cycling between when you found it?

You could TRY hooking up a garden hose and opening that valve, then turning the pump on to flush out the crud that's been stirred up. It's possible-but-not-likely that this MIGHT clear a bit of crud that is jamming the check valve open. If nothing else it will get the stirred-up crud out of the well before the pump is repaired or replaced.

If your well is prone to throwing dirt, you may want to add a "spin-down filter" (I put mine after the relief, but before the rest of the system - this was contrary to the manufacturers instructions - they suggested after the pressure tank - but in line with the whole point of having it.)