An airplane or aeroplane (informally plane) is a powered, fixed-wing aircraft that is propelled forward by thrust from a jet engine, propeller or rocket engine.
From the tag info page for airplane:
A heavier-than-air fixed-wing aircraft propelled by one or more engines.
So fixed-wing is the more generic term, which applies to both airplanes and sailplanes. Therefore, fixed-wing is not a synonym of either.
We might however want to add a more detailed tag info to fixed-wing, which currently only has an excerpt, to clarify the difference.
Regarding the usage on this site: I would recommend being specific (i.e. using airplane or sailplane) whenever possible, e.g. this question:
Why are propeller driven aircraft commonly built with a higher wing aspect ratio and a straighter wing angle?
is currently tagged fixed-wing, but should really use airplane because it is clearly about powered aircraft. But questions, where it makes no difference, should be tagged fixed-wing like this one:
Why do Wingtip Vortices affect the upwash ahead of the aerofoil?
Here, it makes absolutely no difference whether or not the aircraft is powered or not.