High school is considered primary education here. (at least by me

Ah, right. So, this terminology is actually pretty standardised and high school is secondary education, with primary school naturally being…primary education. Though there is a little ambiguity around the edges, for example:

United States: High school (usually grades 9–12 but sometimes 10–12…) is always considered secondary education; junior high school or intermediate school or middle school (6–8…or other variations) are sometimes considered secondary education.

It’s of particular note because the phrase “tertiary education” universally refers to post-school education like university or technical colleges, and it wouldn’t make sense to have tertiary education if there was no secondary.