Since your definition of birthday just said "1+…", I assume you're not talking about transfinite games. In this context, the birthday is the longest a game could go on (in number of moves), and the quasi-birthday is the shortest a game could go on.

It's notationally convenient to treat a short impartial game as the set of its options. Here are some basic facts: The quasi-birthday is $1$ exactly when there's a move to {}. The quasi-birthday of any game with {{}} or {{},{{}}} as an option but not {} is $2$. The quasi-birthday of {{{{}}}} is $3$, and the only games with birthday equal to quasi-birthday must look like that.

The concept of birthday is useful for giving bounds on things in a variety of theorems and proofs. One example is the theorem that if a short partizan game $G$ has finite order and birthday $n$, then $2^n\cdot G=0$. Another example is Mesdal and Ottaway's original proof that in the partizan misère context, $0$ is not equal to any other game. If $H$ is a game, then their witness to the fact that $H\ne0$ is a game $G$ where $G^{RR}$ is $n$ consecutive moves for Left, where $n$ is one more than the birthday of $H$. That way they can use the fact that $H$ will be exhausted of moves before Left can be done moving in $G$.

I do not recall seeing any argument that used the shortest line of play, as in your quasi-birthday idea, and suspect that it is not very helpful when considering disjunctive sums. As Noah Schweber seemed to be getting at, a consequence of the Sprague-Grundy theory is that every impartial game is equal under normal play to a game of quasi-birthday $0$ or $1$. But perhaps with one of the other, more rarely considered, combinations other than disjunctive sum, it could be helpful.

Since your definition of birthday just said "1+…", I assume you're not talking about transfinite games. In this context, the birthday is the longest a game could go on (in number of moves), and the quasi-birthday is the shortest a game could go on.

It's notationally convenient to treat a short impartial game as the set of its options. Here are some basic facts: The quasi-birthday is $1$ exactly when there's a move to {}. The quasi-birthday of any game with {{}} or {{},{{}}} as an option but not {} is $2$. The quasi-birthday of {{{{}}}} is $3$, and the only games with birthday equal to quasi-birthday must look like that.

The concept of birthday is useful for giving bounds on things in a variety of theorems and proofs. One example is the theorem that if a short partizan game $G$ has finite order and birthday $n$, then $2^n\cdot G=0$. Another example is Mesdal and Ottaway's original proof that in the partizan misère context, $0$ is not equal to any other game. If $H$ is a game, then their witness to the fact that $H\ne0$ is a game $G$ where $G^{RR}$ is $n$ consecutive moves for Left, where $n$ is one more than the birthday of $H$. That way they can use the fact that $H$ will be exhausted of moves before Left can be done moving in $G$. (See "Simplification of Partizan Games in Misère Play" at http://www.integers-ejcnt.org/vol7.html )

I do not recall seeing any argument that used the shortest line of play, as in your quasi-birthday idea, and suspect that it is not very helpful when considering disjunctive sums. As Noah Schweber seemed to be getting at, a consequence of the Sprague-Grundy theory is that every impartial game is equal under normal play to a game of quasi-birthday $0$ or $1$. But perhaps with one of the other, more rarely considered, combinations other than disjunctive sum, it could be helpful.