What is the general name for this particular game mechanic?
This is called first in wins.
See for example this article on game design

First in wins.  Markers placed later are considered to fall behind those placed earlier.  The only way a later player can get ahead is to exceed the previous count. This is generally best if the game has a start player disadvantage.

This is in a broader class of tie-breakers favoring efficiency, which one author categorizes as epsilon tie-breakers:

The efficiency argument and even the game balance factor are a restatement of the epsilon argument addressed above. The game balance point is perhaps a little more subtle, but in essence is the same as the many majority/plurality games which state that the first one in is the default winner of the contest (eg Medieval Merchant, Die Neuen Entdecker, etc): another phrasing of epsilon.

(which I found interesting because of the default formulation of this rule: &quot;first one in&quot;, a form of &quot;first in wins&quot;)