Any time an enemy aircraft was taken out of action (shot down, forced down, destroyed on ground) then the pilot was awarded the credit.
However, to get that credit, there had to be at least one witness to corroborate the claim.
This could be from a fellow Airman, a captured allied Airman or observers on the ground (such as Army or Navy personnel).
If the enemy aircraft was downed by an observation or bomber aircraft, the crew were awarded the credit.
The one caveat, however, is that the enemy aircraft had to be downed anywhere within friendly territory to count. If it was downed behind Allied lines, it was discounted.
Example: if Richthofen had downed May's fighter (instead of being killed himself), it would not have been awarded to his victory total (as several others were also disqualified, particularly in 1915).