I can see why he'd make the wrong conjecture. The actual rule as you state it is bizarre.
There are almost 12.4 full moons per year; that's more than three per season. Every season will have at least three full moons, so the fact that a season does have three is completely unworthy of notice.
The fact that a month has two full moons is worthy of notice, since calendar months are just barely longer than lunar months. But when you do have an unusual month... why would that trigger a special designation on an unrelated full moon that isn't out of the ordinary in any way?
Wikipedia's description does make more sense; notice is not taken of whether a month has two full moons. (Though the condition is necessary and sufficient.) Instead, the "blue moon" is called that by virtue of being the extra full moon that occurs in a full year when that year has 13 full moons instead of 12. ("Folklore assigned a unique name to every full moon of the year.") This had to be either the second or third full moon in the season containing the extra moon (since the first / fourth full moons of that season signal its beginning / end).
There are almost 12.4 full moons per year; that's more than three per season. Every season will have at least three full moons, so the fact that a season does have three is completely unworthy of notice.
The fact that a month has two full moons is worthy of notice, since calendar months are just barely longer than lunar months. But when you do have an unusual month... why would that trigger a special designation on an unrelated full moon that isn't out of the ordinary in any way?
Wikipedia's description does make more sense; notice is not taken of whether a month has two full moons. (Though the condition is necessary and sufficient.) Instead, the "blue moon" is called that by virtue of being the extra full moon that occurs in a full year when that year has 13 full moons instead of 12. ("Folklore assigned a unique name to every full moon of the year.") This had to be either the second or third full moon in the season containing the extra moon (since the first / fourth full moons of that season signal its beginning / end).