Is short term disability considered paid family leave?

Yes, short term disability is considered paid family leave. Pay from your employer, a third-party insurer, or the government while you're unable to work due to a serious illness or injury is considered paid family leave .

I am trying to file both long and short term disability forms. How do I correctly do this?

What forms did you get from the employer?  W2's?  Were there wages reported in box 1 on the sick pay W2?  If none, there is nothing to report. Were your premiums paid pre-tax or after tax?  If pre-tax you will pay the taxes on the benefits.  If after-tax premiums then the benefits are tax free.   Here is a link with discussion.  If you believe the payments were omitted from a W2 and should have been taxed here are some steps to enter those in TurboTax.  Otherwise simply enter the W2s that you have.