How to introduce to your new teacher when you have a middle name or more than one different names

My teachers and my friends called me Kitty in my former school

My colleague and my boss call me Kay in office

I am going to learn how to swim, which means I will meet with some new people in the swimming lessons.

I wonder how to introduce myself to the new teacher because I have two different names? Should I say this:

You can call me Kitty or Kay?

and then in the full name row of the registration form, should I write "Kitty Kay myFamilyName" or just "Kitty myFamilyName"?