Move refers to your movement
In the PHB (pp. 190-191), this quote:
Whether a creature is a friend or an enemy, you can't willingly end your move in its space.
appears under the section "Moving Around Other Creatures", which itself is a sub-section under the heading "Movement and Position".
It is arranged this way in the basic rules on D&DBeyond as well.
The paragraph beforehand also uses move in this way:
You can move through a nonhostile creature's space. In contrast, you can move through a hostile creature's space only if the creature is at least two sizes larger or smaller than you.
And the sentence after:
If you leave a hostile creature's reach during your move, you provoke an opportunity attack, as explained later in the section
Therefore, the context for this quote is with regards to movement. Move is referring to using your movement, not a synonym of "turn".