"Light Off" is old school DC9 MD11. I think it has just carried over, however used for completely different reasons. I believe they had lights to indicate ignition. These lights went out at some point.
It is important that one doesn't expect the automation to do everything. We really should be monitoring the automation, and interdicting when necessary in accordance with SOP/AFM. If you are sitting back fat dumb and happy, you are pulling yourself "out of the loop", and as stated, it is possible to exceed 2 minutes of starter operation on a normal start with an abort where high residual EGT is involved prior to start.
It is up to an individual pilot to assess an aircrafts limitation compliance. How you do it is technique and situational awareness. Starting the chrono is one such method.
To say it is distracting is a falsehood. If anything it should improve your situational awareness by reinforcing pertinent points of the start sequence by implementing interactive monitoring, rather than passive monitoring alone. Working with the automation to produce an outcome by following a sequence of events is far more effective as a precursor than just watching.
As pilots, whether it is an auto start sequence or autopilot, we know how important automation awareness is on the Airbus to avoid the "what's it doing now" scenario or automation surprise.