Whether you’re an employee or an independent contractor is a facts and circumstances test. To the extent that an instructor has more control over what you teach, where, and when, they are more like an independent contractor. To the extent that the instructor has less control over these things and the gym has the lion’s share of the say, an instructor appears more like an employee.
It does matter which you are for tax purposes. A contractor fills out Schedule C for their business and takes deductions directly against their business income. Also, they pay both the employer’s and the employee’s share of self employment taxes (FICA / Medicare) so there is a larger payroll tax burden for being an independent contractor. An employee gets a W-2 and although similar expenses are deductible, they fall on Schedule A as an other miscellaneous itemized deduction, and only the amount that exceeds 2% of your adjusted gross income actually reduces your tax burden.
There’s a great publication on irs.gov that tells you about employees versus contractors. That way you and your studio can see what fits best based on your unique situation. Both of you need to be consistent.