OP, I don't know how much you might know about install paths or directories, but you might look at your directory structure for what you have installed. If you download and install "ONLY" from the "Frontier Store", your directories will end with the same "Product" named directories, which is the only thing we are interested in here.
If you downloaded and installed under "Steam" then your directory structure and paths should match what I have listed below.
"X:\path\Steam\SteamApps\common\Elite Dangerous\Products"
"X:\path\Steam\SteamApps\common\Elite Dangerous\Products\FORC-FDEV-D-1010" <(ED1.8 Lives here)
"X:\path\Steam\SteamApps\common\Elite Dangerous\Products\elite-dangerous-64" <(Horizons 64 Bit lives here)
"X:\path\Steam\SteamApps\common\Elite Dangerous\Products\COMBAT_TUTORIAL_DEMO" <(Combat demo lives here)
Now, look at the last directory name in the list, find and go to "\elite-dangerous-64" if you have a 64bit machine (I assume you do) or "elite-dangerous-32" if not, and find the file named "EliteDangerous64.exe" and run it, or click on it to launch the "Horizons" version of the game.
If that works, you do have it installed, and your desktop shortcuts, or the path to the "Horizons" version is not leading to the proper "install directory".
I'm sure you have some "path" problem either in the launch shortcuts, or possibly your install directories are not listed correctly in the registry.
Even if you "uninstall and reinstall", these directory paths are left behind in the registry, and keep being reused until deleted. When I uninstall and reinstall, I delete all previous directories, paths and shortcuts to Elite, and then run a registry cleaner like "CCLeaner", to remove the old directory paths from the registry. Then I reboot, and run "CCLeaner" again, just to make sure, before reinstalling.
Also noted, Is that you should "Pick One Install Type" and stay with it. Either install from the "Frontier Store" or install from "Steam", but don't do both unless your very adept at install points and paths. I wrote a "help post" for those who want to install from Steam and the Frontier Store, which works very well for me, but FDev knocked it down, telling folks to "Choose One" and they don't recommend you install the game from "Both" Install Points.
Here is the link to the original post for more help on this.
https://forums.frontier.co.uk/showt...nd+Steam+only+installation+for+ED1.5+Horizons
Hope you can determine why you can't play Horizons.
Be well.