In the south, 冲涼 is equivalent to shower. This is more like a douse from a pail of water (in the old days) to cool oneself in the tropical humidity. In the north, where it is much colder, would not "wash" themselves in the same way. In ancient times, and to a certain extent, nowadays 沐 means to wash the hair. 浴 is to cleanse oneself in a pool of water. 沐浴 becomes a ritual/formal cleansing before some important event takes place - hence the phrase, 沐浴更衣. No need to be precise about taking a bath or a shower. Context will determine the meaning. For example, in HK, except for grand hotels, there is no space for a bath tub. Even here in Los Angeles, the bathtub is heading the way of dinosaurs. No one has the luxury of spending time in having a bubble bath! Next year, I am going to remove the bathtub and remodel it into a walkin shower like the gym in my house! It is an ass job trying to clean the tub even when it is not used for its intended purpose.

take a shower is more common in CHINA. we call that淋浴，淋is a verb when rain drops on your body.taking a bath is 泡澡，洗澡is what we use the most.That means both of them but especially take a shower.

To make it simple, you use "洗澡" for both when you just want to tell what you will do but not concrete how you will do it. For emphasis detailed way you do it, use "冲澡" for a shower, "泡澡" for a bath.