When I say plane, I always refer to a flat plane. I can intuitively perceive the difference between a flat plane and a curved surface. The fact that a small region on a large curved surface seems flat is just an illusion and should not enter into a rigorous scientific discussion.
A line always refers to a straight line (except where I explicitly call it a curved line in quotes) and a triangle always refers to a polygon made by three straight lines (except where I call it a curved triangle, in which case the sides are assumed to be curved lines).
If I happen to be dealing with a curved surface, then I know that it is a curved surface from my frame of reference. I can still perceive a straight line on a flat plane, living on a curved surface. If I draw a straight line on a flat piece of paper held perfectly horizontal then that line is parallel to a tangent to the surface of the earth.
However, if I choose to redefine a line on a spherical surface as different from a straight line on a plane, then I am again making a conscious decision to do so, and the redefinition changes the earlier definition of the line, and hence, it is to be expected that some of the rules change with that.