I once saw an article and/or heard a shiur from rabbi bar hayyim, calling succot in the diaspora, 9 days.
Why would he consider simchat torah in the diaspora, to be a day of succot?
Or even shmini atzeret.
From what I understand, succot is 7 days in israel and 8 days in the diaspora.
It is stated in tenach that succot is 7 days. Yes, it then says "on the 8th day", but that is not succot. That would be 8th day of festivities.
Mishna Moed is a bit clearer
In the diaspora, we follow the customs of our ancestors, on which, on the day that is the 22 Tishri, there is a safek(doubt) as to whether it is really the 22nd, or whether it is the 21st. So we keep both, but not to the extent that one would detract from the other.
A proof that Shmini atzeret is not succot, is that on succot we have to sit in the succah. in israel, where we follow the custom of where there is no doubt, where we know that "22nd tishri is 22nd tishri". *Nobody* sits in the succah.
That is because Shmini Atzeret is not Sukkot (btw, and vice versa - SK is not SA).
Whereas in the diaspora, some do sit some don't sit.
They are not just "independent" and "distinct". But one is not the other.
In the diaspora, it is not even an overlap of succot and SA,it is a (custom of a) doubt as to which it is.
thanks
Note- From what I have heard, nobody sits in the succah. Is that correct? If not, then it would break some of my argument. But I think there is something in the talmud to back that up, about in the diaspora whether you sit in the succa or not, because of the doubt. Suggesting that in israel you don't.