What happens if you try and keep it running by giving gas? Will it stay running?
If the security light isn't flashing, its not the transponder.
Seems like a fueling issue. When was the last time you changed the fuel filter?
Do the test again like you did in this video, but this time dont give the fuel pump a chance to prime. At 0:20 in you turned the key on and left it for a few seconds, this gave the pump a chance to prime and it started again.
Start the car and let it die then immediately start it again without delay. If it doesn't start the second time then it is a fueling issue (pump, clogged filter or lines).
When you turn the key on without starting can you hear the pump priming? How does it sound? It should be fairly quiet.
Testing your fuel pressure is a good easy next step.
I suggest upgrading the pump anyway. Its only $80 and really makes a difference.