Some cars have running lights (which are on whenever the engine is running). Some states have laws the require headlights be on when wipers are being used. Do running lights meet that requirement?
No, they don't.
A vehicle, whether stationary or in motion, on a public way, shall have attached to it headlights and taillights which shall be turned on by the vehicle operator and so displayed as to be visible from the front and rear during the period of 1/2 hour after sunset to 1/2 hour before sunrise; provided, however, that such headlights and taillights shall be turned on by the vehicle operator at all other times when, due to insufficient light or unfavorable atmospheric conditions, visibility is reduced such that persons or vehicles on the roadway are not clearly discernible at a distance of 500 feet or when the vehicle's windshield wipers are needed [...]
On all cars I know of, having running lights on does not turn on the taillights, so using running lights would not comply with this law.
This point is also emphasized in this MassDOT press release, which, while it is not law, does give some indication of how the government intends the law to be interpreted:
Relying on daytime running lights for these conditions is not sufficient under the law.