Am I really self employed?

You may be considered a contractor for the hair washes, but the rest is general employment. Sister has no idea what she is doing, and could end up in legal trouble with the state labor board. And, you could cause that trouble by asking this question to the people you should be talking to, which is your state's Labor board.

Your sister (and her partner) need to learn how to legally run a business.

You need to consult with a tax consultant in your state. The setup sounds fishy to me. MAYBE you can file a 1099, but at this point I think you should be given a W2 form.

For discussion of the difference between the two terms, please google "employee or independent contractor".

It happened to me; apparently the corporation I worked for had a legal basis for doing it.  I didn't analyze why employers use this popular if irregular trick, at the time or since then.  You will get good info here at Yahoo Answers, I'm sure.  More important than "why" is what you have to do regarding Taxpaying and possibly Labor protocols to get onto the legal track.

You can file a Form SS-8 to ask the government to decide how you should be classified.

If your sister doesn't list you as an employee, doesn't pay you for all of your work, and isn't withholding taxes for you, you are not her employee.  Should you be? Yes, yes, you should, but if you didn't fill out employee paperwork or a a W-4 tax withholding form, you aren't an employee of her business right now.

Since you agree to this and get paid by stylists in the salon, you are basically an independent contractor.  You would have to file your own taxes anyway, so this isn't that different.  The big issues are that you haven't been paying into social security or state or federal tax, so you may own more, and how you account for your income if you haven't been treating your job as a self-employed contractor that invoices. Hopefully you have some record. You'd be wise to talk to a tax accountant.  

Classification of employees is typically handled at the state level.  California, for example, is much more strict about classifying people as employees than most other states.  

If you are self-employed, typically you set your own hours and negotiate your own wages while being free to provide their services to multiple companies.