And then there are Blue Moons - I remember a Summer in the eighties, I think in July, when this happened; we could see them both here, and it seemed odd and wonderful to me at the time:
https://en.wikipedia.org/wiki/Blue_moon
OK, then the 13th moon in August could be Blue Sturgeon, but in September, the only possibility would be September 1 and 30, but September 30 would be Harvest Moon, with the September 1 moon unnamed.