That IS the starter relay, and it's important to understand how they work, simply
The "push on" terminals at the top are interchangeable and feed the relay coil. One is the "start" signal from the key, the other is grounded when the trans is in park/ neutral.
The "big stud" is battery, and one contact of the relay
The other contact of the relay is the "square" terminal and feeds the starter.
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First thing to do is take a screwdriver/ other metallic tool and jumper across the battery stud and the square terminal. MAKE SURE the transmission is in park/ neutral. If the starter spins, you know the battery, cables, starter are OK
If not, move your screwdriver down to the starter and jumper the two terminals on the starter. If you still get nothing, the problem is
bad battery clamps/ cables
bad/ discharged battery
bad starter
If the starter spins when jumpering the starter but NOT (confirm) up at the relay, then it's simply the wire between the two.
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NOW IF in the first test, the starter spins, now we have to determine if the trouble is:
the starter relay
the neutral switch
the bulkhead connector
or the ignition switch.
So identify which wire goes down the firewall, past the engine, and down to the transmission. Pull that wire off the relay, and clip lead that relay terminal to ground
Twist the key, test for starter. If it cranks, the wiring, neutral safety switch, or transmission linkage.
If nothing, then...............
Put a test lamp or multimeter on the remaining push--on terminal--the one you have not disturbed
Twist the key to start. If you get battery voltage, the relay is bad
If not, either bad connection in bulkhead connector or bad ignition switch