Hi Martin,
What you would have to establish is whether the off day in the original Red Sox schedule showed them off on either May 9th or May 10th, 1912. Perhaps someone on net54 has a 1912 Red Sox schedule? It might be worth asking the Q.
The ticket you own is a stub, so it is obvious that a patron used it for admission at some point in 1912. Since May 8th (the date on your ticket) was a rainout, it would have been used to gain admission for the
unscheduled game on either May 9th or 10th, 1912. There would not have been a pre-printed ticket to this game with the date on it.
I say this because of the info that you provided about the Red Sox were hosting Detroit for a 3 game series and they played a 3 game series in spite of the rainout.
As to your Q as whether your ticket was used for the 2nd game of the series on May 9th or 10th: It would depend if the 2nd game played in the series was the unscheduled one or a scheduled one. If May 9th was the unscheduled date, you may have a possible case.
Also consider looking at the original Boston (Globe) newspaper archives online. You may find a clue as far as what actually happened within these game recaps, especially because it's a local paper.
Scott