I may have missed this during Ticket Support, but at any time did you perform all of the following:
1. Install the Lite version or any earlier non-Lite version.
2. Have trouble uninstalling that version.
3. Have someone not-from-VaultWiki help (or yourself) uninstall that version by modifying your code or database.
In this case, what may have happened is that:
1. The helper removed the add-on's internal uninstall function or the call to it in order to skip that code, because there was an error.
2. VaultWiki (Lite) appeared to be removed successfully, but actually all wiki tables and data still existed.
3. Later you try to re-install VaultWiki and it detects the existing installation, upgrades it. However, the add-on entry is still missing because that was uninstalled in step 1. Phrases templates, settings, navigation, etc fail to update because the associated add-on entry does not exist.
4. You have the result you saw.
We were able to fix this on your site by tricking VaultWiki into thinking that it wasn't installed yet. This ran over all the original install steps and created anything that was missing (like the add-on entry).
However, I cannot be sure now that if this was the case, that the database is currently in a usable state. If you encounter any database errors complaining about missing columns, then we still have some more work to do fixing this on your site.
If this process is true, then this is not actually a bug in VaultWiki but was caused by end-user modification.
I looked into the possibility of double-clicking being a cause and found that double-clicking will only ever cause the same step to occur twice or to run the next step in the background. In both cases, the step number increases as expected and no step is skipped. The only ways to "skip" a step then would be to manually type the step into the URL (which I doubt you did), or if there was already a step number in the logs from a previous installation (which is how I came up with the above theory). In either case, VaultWiki is working as designed.
While there are ways to avoid the case in my theory by using some kind of unique hashes if you are trying to re-install VaultWiki from different versions or at least different downloads, I don't know if the effort to develop such a solution is worth it, since it requires the end user to corrupt the previous installation manually and that is such an unlikely case.