Load the Student VectorCalculus package and define the vector q:=Vector(<1,0,0>).
Change to spherical coordinates:
MapToBasis(q,spherical) => (1)*e[r]+(Pi/2)*e[phi]+(0)*e[theta]
So, the Physics convention has been imposed, as it has been since at least Map;le 2016 (thee earliest version I can access). To reverse the meaning of the names phi and theta, do MapToBasis(q,spherical[r,theta,phi]) =>(1)*e[r]+(Pi/2)*e[theta]+(0)*e[phi]. Note that it's the position of the angle names in the triple of coordinate names that determines the meaning of these angle names.
However, MapToBasis(q,spherical_math) leads to an error (spherical_math is not one of the 5 coordinate systems supported by the Student package). Therefor try
VectorCalculus:-MapToBasis(q,spherical_math) => (1)*e[r]+(Pi/2)*e[theta]+(0)*e[phi], which is the same as if "spherical_math" were just "spherical."
The reason for this: the "changecoords" command applies only to scalars. To change coordinates in a vector in one of the VC packages, the MapToBasis command is essential. Otherwise, the basis vectors will not be subjected to the change.
Also, note again that in the VC packages, it's not the names of the spherical angles that matter. Perhaps for display purposes, yes. But the function (i.e., meaning) of the angles, no matter what their names, it's the position of the angles in the triple [name1, name2, name3]. The middle name in that triple is always the angle down from the z-axis; the third name always refers to the angle around the z-axis.
Perhaps the help page at ?coords could have mentioned these two issues - (1) the changecoords command is a top-level command, and has nothing to do with what happens in the VC packages. In those packages, the MapToBasis command is much more important but has been left ignorant of the option spherical_math; and (2) the meaning of the names used in spherical coordinates is determined by the position in the triple of coordinate names, not by the names themselves.