Yes, it was individual tickets punched by tellers at specialized
(by type and amount) bet windows. I believe the accounting was
done in the background by machines, though. The name 'American
Totalizer' (?) comes to mind ('tote' for short)
as the primary supplier of that equipment. Of course, there was
at least twenty minutes between games to allow for the betting
activity. I always used a telewager girl since you never got shut
out - they did not close the mutual windows until the girls had
phoned in all their bets.
Re those girls: they worked a specific section and since I always
sat in the same section I always had the same girl. If she was
a working girl then she was a middle aged working girl and I never
knew it. What I heard was that they did away with them because
they were holding bets out and making book themselves.
I am the flip side of the original poster - have never used any
automated betting machine since I've not attended a live performance
in almost thirty years!