What you think of as "Greek" style yogurt is literally just normal yogurt run through cheese cloth or another similar filter (at home, I assume industry has a much more efficient method) to allow liquid to drain out.
The difference in the process is simply that once you're done making "normal" yogurt, you strain it to remove some liquid whey.
You can take this to varying degrees. While I was still dialing in my process, I made some yogurt so dry I questioned whether it was actually cheese. It made you thirsty to eat it. I've since reduced the time I spend straining it to something a little more moderate so it's still kind of creamy but not wet or runny.
The FAGE-Total, wherever I find it, it costs around £3, while the "Greek type" costs £1 is is significantly more muddy/wet. I can taste the differce in quality, and they definitely make for a different outcome in any sauce you are trying to make.
Semi liquid sounds excessive for actual yogurt, but what you may be looking for specifically is strained yogurts, which have a lower amount of whey (liquids) and are thus drier and thicker. Greek yogurt is indeed a common type of strained yogurt.
I have eaten yogurt all over the world. The "Greek" yogurt (or in some countries they call it "Greek Type" - to avoid 'lying' about the country of production/origin) is thicker/less of a liquid. It must be in the process. Most yogurts (plain, not sugary with fruites and other stuff) I've had are semi-liquid. The only 'good' yogurt I truly enjoy is the original Greek by "FAGE" called "Total" (not affiliated).
I assume that similar yogurt-making process must be a regional thing (Greece, Balkans), and perhaps Greece had the industries/factories to support exports on great volumes that helped them conquer/define this type of yogurt in the 80's, 90's and establish that name.