Yup that’s a starting relay. Typically those are used to put more current to the starter solenoid “start” connection, without that extra current going thru the ignition switch. But it looks like they also used its post for some other electrical connections too. Without seeing your car’s wiring diagram, it would be hard to say for sure. But I would guess that the upper (thinner) wire on this relay probably goes to the starter solenoid’s “start” terminal. You can check it out when you pull the starter with an ohm meter. Or perhaps you’d like to see what the voltage is at that top terminal when the starter is trying to run. But if you’ve tried jumping (eg shorting the two big terminals on the relay) and that did not change the starting behavior (it should have made the starter try to go without the key being at start), that’s probably as good a test.
Oh, the top big post on the relay probably isn’t ground. It likely gets connected to the bottom big post (battery and alt) when the start signal comes in, so if it were ground then you’d have one heck of a short. It may measure close to ground cause it probably goes to the start terminal on the solenoid, which is pretty low resistance (<1ohm), so might look like ground but probably isn’t. I try to use a clean spot on the engine as my ground reference. That is a reliable ground. I think if you measure voltage between that top terminal and the engine when the key is in start, you’ll see ~12v.
One other test which is fairly quick, put your hand on the relay and have someone turn the key to start. You should feel a click as the relay operates. And if you can get to the solenoid on the starter do the same there. Again you should feel a click as those electro magnets energize. If not whichever isn’t clicking is where the problem starts (may be the wiring to or the unit itself).
Tranny interlock could be involved. But again, if you can jump the starter terminal (the small one, which likely goes to the relay) that would get around any issues with the tranny switch, ignition switch, relay, etc. You might also try starting in N instead of P, sometimes that will make a diff.
If you pull the starter (be sure to disconnect the batt first) take a close look at the wires and connections there. If any are dirty/corroded, fix them first and retest before pulling, you might save some time. You can do a first level test at home, but that really isn’t the same as under load. I’ve had starters that would seem just fine on the garage floor, that were bad and failed under load when tested.