Why are there multiple ways to express "Wait a second" such as 等、等下、等一下？
Why not?

English has the similar phenomenon. For example, the following phrases have the same meaning:
wait a bit
wait a mo
wait a moment
wait a second
wait a minute
wait a jiffy
wait a jiff

The following phrases have the same meaning:
等一下
稍等一下
稍等一等
稍等一會兒

等下 is not so common.

Usually, 等 means "to wait", not "to wait a second".
For example, 等人 means to wait for somebody.
It is different from "to wait a second".

"Wait" can mean "wait a second" in English.
等 is not the same as 等一下 in Chinese.