"I am on my home" is correct. You are right - for any other destination we use "to". For example "I am on my way to the shop", "I am on my way to work", "I am on my way to your house". But "home" is different! Why? I don't know! But it does not take the preposition 'to'. For example, compare these correct sentences "Let's go the the cinema / Let's go home", "I got to the office late last night got home late last night", "It's a long way to London from here / It's a long way home from here"

Well of course it would be "I am on my way home."

They're both right because they're the same

The second one

I am on my way home. :D