If that's a new battery and the correct size check it again with voltmeter while cranking. It sounds/looks like a weak battery, watch the drive shaft, the solenoid/relay should keep the starter engaged whether the engine turns over or not. If you have close to 12v when start button is pressed then you have a high resistance connection or even a corroded battery cable possible under the insulation, if it's a cable it will be hot where the high resistance is. Seriously, watching your video closely you can see the drive shaft moving a little then back, then move again just like when you have a battery so weak that when you press start it will engage for a sec and as soon as it engages the starter pulls amps which causes a voltage drop, a weak battery pulling amps will cause voltage to drop so far the solenoid/relay will not have enough voltage to keep the relay energized, as soon as it drops connection voltage returns as no amp draw exists, relay connects and starts the cycle again. The faster (lower) the volts drop the faster the clicking. As a side note, if battery is actually good and you hit the start and it stays connected but doesn't turn over, STOP, fix whatever is preventing rotation (hydrolocked, crank/counter bal froze, whatever) if you keep starter engaged and no engine rotation you will stall burn the starter commutator, what the brushes make contact with, then it could give you the symptom that you have now because a stall burned com can be carbon fouled and/or pitted causing, again, a high resistance connection. I rewound locomotive 3k lb traction motor armatures, rebuilt the frames with fields and interpoles, assembled and test ran them among lot's of other work for the Santa friggin Fe RR, same direct current tech as starters so not blowing DC ionized smoke up your chimney. Lee