Trying to identify cause of minor leak from water heater TPR valve
It is fairly common for water heater relief valves to start leaking after you have manually opened/closed them. Sometimes it's due to boiler compounds on the seal/seat surface(s), sometimes due to seal/seat wear, damage, or failure.

I would not assume that it has failed, in that it would not open during an over-pressure/temp. event; however, it may continue to drip/leak which may be unacceptable to you.

The fact that it opened when subjected to the shock wave caused by the closing of the washing machine valve seems to indicate that it is functioning, albeit maybe it opened a bit prematurely because it's partially unseated since you "exercised" it. Then again, with the expansion tank operation potentially compromised, maybe the relief valve did it's exact job perfectly.

In any case, relief valve replacement is an easy job and they are relatively inexpensive. It makes sense to go ahead and replace, for your safety.