It is important to notice all the parties titled Independent Party.
If you join any party, you are no longer independent as far as voting goes.
Look for 'unaffiliated' or 'No Party' or something like that when you register.

There are many voters who don't understand that.

Are you an independent voter? You aren't if you checked this box
The American Independent Party is California’s largest third party. A poll shows 73% may be in it by mistake. Are you one of them?
Independence Party’s confusing name has tricked thousands of New Yorkers

I only found examples for CA and New York. Surely this problem exists in some other states?