Is there any problem in naming "ghusl" as "Istihmam" (bathing), especially when using common terms like "tatassabah" (you are bathing), "tasabbaht" (I bathed), or "na'eeman" (blessings upon you after bathing)?

There is no harm in calling ghusl "istihmam" (showering/bathing), for the word "istihmam" is synonymous with "ightisal" (washing/bathing) in the Arabic language. Istihmam is bathing with any water whatsoever. In the Musnad, it is narrated on the authority of Ibn Abbas that one of the wives of the Prophet, peace and blessings be upon him, took a bath (istahamat) from janabah (ritual impurity). Then the Prophet, peace and blessings be upon him, came to perform ablution from the leftover water, and she said: "I bathed (ightasaltu) from it." He said: "Indeed, water is not defiled by anything."