Actually come to think of it, lexus just tested the battery and said it was good when i was there 2 weeks ago.

 

Personally I'd say your battery is fine.
In the video when you press the Start button the second time, it cranks normal, and strong, and the vehicle starts and runs.
It's not laboring to crank and the crank time was no longer than expected.

Now...when it shuts off that first time, that certainly seems odd.
When cranking the dash lights should all go out so that max power is diverted to the starter, and of course once the engine starts, power is restored to the instruments.
When it cranked the first time, the dash lights went out, as expected, but then it stops cranking...my first guess would be a starter or starter circuit issue.
These cars aren't known for bad starters, so it's a little puzzling.
Unfortunately the car is new to you, so it's hard to have an accurate history of the vehicle.
Could it be possible this thing has an aftermarket remote start that you're not aware of?

 

There is that possibility that the battery could have a bad cell.
Meaning that at times the battery is perfectly fine, and will test fine as well, but as soon as the cell acts up, the battery basically goes open circuit, therefore killing all power to the vehicle.
A bad cell act up or return to normal in the blink of an eye, so it's a little challenging to catch it sometimes.
Is it a Lexus battery or something else?

 

Appears to be the factory battery and nothing aftermarket installed except a k&n intake by previous owner which I want to replace with factory intake if I can find one

 

I'll try and get another video of the entire thing tonight