Joseph: False! In the world of tuma and tahara being kept on the level of eating teruma and kodshim, there is one of the 18 gezeiros in the first perek of Shabbos about showering causing tuma. That does not apply to our level of tahara and to our mikvaos of today. Especially men’s obligation. And when it did apply, it included every shower one would take, even in the privacy of his own home – not related to the mikva.
There is a legend that someone asked RSZA zt’l if he should shower before or after the mikva. He said before because of bain adam lachaveiro, and then again after, because not everyone is makpid on bain adam lachaveiro.