Why the unix bin directory named in that way?

To my mind, bin means something more along the lines of "executable" than strictly "binary". (In the end, all files are binary.) There are quite a few script files even in /bin; on my Debian system, file /bin/* | grep -c 'shell script' says there are 19 shell scripts right there. Sitting in /usr/bin I have another 325 shell scripts.

If your script is usable as if it was a compiled application, just drop it in an appropriate bin directory, as that's where people would expect to find it. If it is a simple helper script, it may be better to put it in a separate directory, just to emphasize that it isn't really an application of its own.

In the earlier times of Unix, the /bin directory was only containing compiled binary files. Scripts came there later. At least from version 7 which introduced the #! interpreter convention but probably later.