Do we know the exact date of the first commercial launch? I.e. excluding any test launches.
We currently do not know the first date of the first commercial launch, other than that they have fast tracked their testing program.
#StillTesting, their 2nd test launch, however does include a number of commercial payloads onboard, including 2 Dove satellites for Planet, and 2 Lemur-2 satellites from Spire, and launched successfully on 21 January 2018.
Whether this qualifies as a commercial launch is up to you — future launches may be "all commercial" but will absolutely include testing elements, as RocketLab models its agile methodology after bigger newspace companies such as SpaceX. So they will always be testing. They'll just be making money doing it :)