You are really introducing your name, not yourself. In a formal situation I would go with either of these:

Hello, my name is FirstName LastName, but you can call me Kitty...
Hello, my name is FirstName LastName, but (most) people call me Kitty...

While I think "I am" is commonly used, this is more suitable for describing your person. You could say something like this:

Hello, my name is FirstName LastName, but you can call me Kitty. I am a doctor with 15 years experience in...

Informally I think you could go with either my name is or I am.

BTW, I would not use "Hi" in a formal situation. Hello or I am pleased to meet you would be better.
In fact, if you start with I am pleased to meet you, then my name is follows naturally:

I am (I'm) pleased to meet you. My name is FirstName LastName, but you can call me Kitty...

You are really giving a salutation and informing your name, but you are not saying anything else about yourself, yet.
In a formal situation I would go with either of these:

Hello, my name is FirstName LastName, but you can call me Kitty...
Hello, my name is FirstName LastName, but (most) people call me Kitty...

While I think "I am" is commonly used, this is more suitable for describing your person. You could say something like this:

Hello, my name is FirstName LastName, but you can call me Kitty. I am a doctor with 15 years experience in...

Informally I think you could go with either my name is or I am.

BTW, I would not use "Hi" in a formal situation. Hello or I am pleased to meet you would be better.
In fact, if you start with I am pleased to meet you, then my name is follows naturally:

I am (I'm) pleased to meet you. My name is FirstName LastName, but you can call me Kitty...