Word 2016 has two personal template locations.

One, which shows up under File > New under Custom or Personal labels. That one is the default save location for templates. It is set under File > Options > Save.

The other is the location of the normal template. That is the User Templates location. That location is set under File > Options > Advanced. This is where Word 2003 looked for templates "On my Computer."

Both of these locations can be changed. For most personal users, they probably should be the same location; they are not in the same location by default.

See File New Variations in the Versions of Word.