Gothic subculture in 2014 is just a punch line that middle aged people use to talk about questionable fashion choices they made in the 90s. And yes, Gothic music was a part of it and probably less embarrassing than the clubs or clothes of the era. Otherwise, it would have been something else.
Gothic rock music is a primary influence on gothic subculture, although the music itself is a reflection on the subculture to begin with. The gothic subculture that spawned somewhere around the early 80s was indeed part of the musical subculture of British post-punk that it originated from. As it is, music is the biggest influence on the subculture as it still stands.
Yes, music that is labeled as Gothic rock is a primary influence on Gothic subculture, because that is the target audience where the music is aimed. Gothic rock is intended to be a reflection of the Gothic subculture, just as all music is intended for the subculture at which it is aimed. It is no different than Disney writing for children.
I do not believe music that is labeled as Gothic rock is a primary influence on Gothic subculture. The Gothic subculture has been around for years and people tend to float into it and then float out of it. I believe the music labeled as Gothic can influence these people, but it is probably equally influenced by the people in the subculture. It goes both ways.
While there is no true genre of "Gothic music", there is a type of music that has been labeled as Gothic rock. Generally, it is rock music that embodies the dark, gloomy aesthetic of the goth lifestyle. I think that gothic rock has been more influenced by goth than the other way around.