I looked at the manual and the call button blinking 6 times indicates a:
Lower Terminal Limit Fault
Lower terminal limit activated.
I assume this means the car has moved too low in the shaft? When it was working the pump would kick in every hour because the car would slowly drift down, not sure if that is normal. How can I force it to move the car back up? There is a control panel with a lot of wires and a switch that says run/off/manual and a reset button. There is also what looks to me like some breaker switches. I can get a picture later if needed.
Boa is offering you sound advice. Your elevator is in need of repair. Call several local companies and ask them what they would charge to diagnose your equipment. Give the same information you provided here.
The safety of you and your family is our only concern. There are many hazards to be aware of when working on an elevator. And if you look through the posts in the section called "SAFETY TALK" on this forum you can see that elevators feel no remorse.
The way this elevator seems to work is there is a hydraulic pump to move the elevator up and a bleed valve to bring it back down. So would a technician would manually turn on the hydraulic pump. Is there is a switch manually turn on the pump? Would a technician have to jumper a relay?
A good valve should not leak/down drift.
Do not make it worse trying things yourself, you are correct, there is a very small pit on this = NO REFUGE SPACE. If this is a sudden loss of hydraulic pressure when you are under it there is a break that will stop it, but it only needs to come down a foot to smash your skull or if your on top of it to throw you off the side of the car.
There are methods of running it up but that is not going to help you. If a Hydraulic elevator drifts it is certainly a leak somewhere, the piston packing seal, the piping or worse, in the valves itself. You'll never see it since it will drain back into the tank. It will keep happening until the root cause is found, that is for sure.
If you mess around and try to get this to pump on its own and burn out the valve body, I can tell you right now you are looking at a $2k repair and waiting 4 weeks (If you are lucky) for the part. O' rings and packing seals are easy but you're going to have to get a company out there.