My name is Amir Khan but instead of Amir I want use Am Khan on my email accounts, online presence, and that's what I want people to call me by and on everything legal I will of course use my full legal name.
Now, when I introduce myself to people I want to say "Amir, but you can call me Am."
Is that going to look professional or is it fine? Am I over thinking myself?
I suspect you're overthinking it :-) I guess these things are always culturally-dependent (you don't say where you're based) but from a UK perspective, I'd certainly think nothing of it. I've known Jonathans who used "Jon", a Thomas who used Tom, a Nikhil who went by Nik and (by way of further variation) a Ketan who used Kevin. Some of these were in quite formal environments, and no-one thought it even slightly unprofessional.
If all of your friends call you something besides your full first name, and you're happy having your business contacts call you that, nothing should stop you from having them use it. As Giles points out, using shortened names or variations on names is both common and acceptable. In your specific situation, I don't think anybody is going to complain about you using Am instead of Amir. If anything, they'll probably feel like they're "in the club", because they're calling you the same thing your friends call you.
Also worthy of mention here is that your name is a part of your personal brand (which may or may not be directly tied to your business's brand). Different names may convey different things to people, and it's worth considering what impact you're going for, and what impact your various name options will have. In this sense, it's not terribly different from choosing a company or product name, except that your parents did the hard work of establishing a baseline.
As an example, James, Jim, Jimmy, and Captain Hook all have very different feels to them, and depending on the level of formality, friendliness, etc. you want to convey, you may choose one over the other. In fact, it's entirely common for a person to go by different names in different social settings.
Feel free to choose whatever name helps your convey what you want to the people you're working with.