Certainly the NYNH&H D.C. equipment had a great deal in common with the interurbans. This is also true of the New York Central electrification out of GCT which is mechanically similar though more advanced. The differences are in wheel profile, size, weight and speed. Those huge cars could not have run on any interurban at that time. Connecticut Company cars, with compromise wheels, could and did run on some of the NH DC lines but, the New Haven's own electrics did not venture onto street railway track. the one known attempt, to run PW&B cars into downtown Providence, was a dismal failure, due to the full MCB wheel profile, and eventually led to the building of the tunnel and viaduct to the station as the only way to reach downtown. The differences were similar to the differences between Rapid Transit and surface cars though, the New Haven cars, some of which were actually converted steam coaches, were even bigger.
The Branford article gives the impression that the Nantasket Branch once had third rail but,
that was only on the Braintree-Cohasset section (and in Conn.).
As for Hartford-Bristol being the first permanent installation, at the time the company believed that Braintree-Cohasset was the first stage of a comprehensive electrification of the Boston suburban system and it was as serious an electrification as could be done at the time. South Station Boston was very congested and the operating characteristics of MU trains would have greatly increased capacity. Local objection to the exposed third rail and change of management objectives changed that and longer trains rather than increased frequency solved the problem until the decline set in. Though in 1907 South Station was handling over 1900 moves a day which was, at the time, a world record.