Add a GFCI protected outdoor outlet
Would it be GFCI protected?


It would be if you install it properly, by splicing off the load side of the GFCI in the garage, or off of the light switch or the 'other outlet' you mentioned.


  Does an outdoor outlet need a GFCI plug?


No. It needs a receptacle (not a plug).

That receptacle should have GFCI protection.  That requirement does not mean you have to install an outlet with GFCI buttons outside.  In fact that's a bad idea because those buttons won't weather well.

But if you provide power to the new outside outlet VIA splicing into the load side of an indoor GFCI outlet, or GFCI-protected circuit, a plain old cheap outlet will be perfectly GFCI protected outside.