I had the same problem recently. I ended up reinstalling VS Code.
Do I lose my settings?
No. Note that I did not uninstall VS Code, I just downloaded the installer from the website and installed VSCode. This won't hurt, as it does not touch your settings and extensions.
What I tried
I first tried to visit the path mentioned in the error log, and I noticed that the folder _ within that directory contained the new VS Code version, downloaded by the updater.
So the path should have been C:\Users\myusername\AppData\Local\Programs\Microsoft VS Code\_\Code.exe instead, perhaps.
I tried to move all files from _ to their parent directory, and then, in VS Code, Help » Restart to update. This opened the new version of VS Code, but after closing it and starting it from the original install location (or through a shortcut I have always been using), the old version was still loaded. So apparently, this "trick" didn't work.
Then I tried to make a symbolic link to Code.exe within the _ directory, so the VS Code updater would find Code.exe. This caused roughly the same things to happen. So, this didn't work either.
At last, I decided to just download VS Code from their website, and simply install it. This seemed the most simple option to me.