From my files ...
$1, 2, 3, ...$ are cardinals, they count as in $9$ trucks, $12$ voles, etc.
First ($1^{st}$), Second ($2^{nd}$), etc. are ordinals, they're used to order stuff.
Now, I've heard of the $0^{th}$ law of thermodynamics, but even the late great Isaac Newton stated his $3$ laws as $1^{st}, 2^{nd}, 3^{rd}$.
In the world of sports, it's the same: $1^{st}, 2^{nd}, 3^{rd}$.
Even in math, with sequences (arithmetic/geometric/etc.), we speak of $T_1$ (first term), $T_2$ (second term), so on.
Why is it, o mathematician, that $0$ makes less/$0$ sense as an ordinal than as a cardinal?
I suppose we could say that ordinals have existential import i.e. it implies esse (being) while cardinals do not. $0$ is associated/mapped with/to nonbeing or nonexistence.
We do treat $0$ as an ordinal when appropriate. Think about the terms in a power series indexed by their degree. The $0$th term is the constant term.
In computer science the most useful default for arrays, vectors, sequences is to start with the $0$th term. So a sequence of (cardinal) length $4$ has terms up to the (ordinal) $3$rd.