What is the spec of your NSX?
From the gauge design, looked to be DBW so must be 95+ MT.
Year model, country spec, any modifications, aftermarket security devices, etc?
Anything done around the ECU connectors, behind the seat, ICE devices, etc?
Anything done inside the Eng bay when this started happening? For example, DBW motor connector, sensor connectors, etc?
Please read the error code first.
http://www.nsxcb.co.uk/showthread.php?9229-How-to-read-the-error-code
If you set the clock, does it keep the correct time?
In your video, after in P2 ON, the CEL stayed On for about 2sec for the bulb check mode (normal) but it immediately came back On again (00:05 - 00:06) and stayed On for at least another 2+sec (not normal).
Possibly triggered the CEL before starting the engine due to open/short circuit on sensor/device/etc or loss of Vcc or Vref.
It actually stayed On for about 3sec but then you cranked the engine so it could have stayed On forever (not normal).
The engine stalled fairly fast so not conclusive but your TCS light also stayed On after the engine started.
Normally, it will switch itself Off as soon as the engine starts.
However, because it stalled so fast that it may not be able to complete the cycle and could be nothing wrong and not related.
Just before the 2nd attempt when you turned the IGSW into P2 ON, the CEL DID switch itself Off after the 2sec bulb check mode and stayed Off (00:16 - 00:19, this is how it should be).
Then, the engine started successfully and the TCS light switched Off.
From what you wrote, there are several question marks on the IGSW and you could have multiple issues resulting in just one phenomenon so another question mark (but lower priority than the IGSW) on the Fuel Pump Relay, Fuel pump, fuel pump resistor but I'll wait for your answer to the spec of your NSX and the report on the error code first.
Kaz