First off, as long as you have a backup of the file there is certainly nothing to worry about.
The update command above looks fine, but I think you can actually skip the backslashes in front of the spaces there. In any case, instead of trying to figure out what exactly went wrong, I’d suggest to start again by making a new copy of your backup copy and working with that instead. Here are some observations which might help:
The ‘no such’ table error suggests that you didn’t open the actual project file. Note that you can open any file with the sqlite3 path/to/anywhere.tpy command: if the file does not exist, it will be initialized as an empty SQLite database. If you then run a query on the photos table, it would rightly tell you that it does not exist. So, this probably just means that you mistyped the file name or that you were in wrong working directory.
The update command you posted above would not change the path from Downloads to Desktop, so I suspect that some of your files are currently linked to the Downloads folder and some of them to Desktop folder. If that’s the case, you will have to run two separate update commands, like you tried.
Like I said, I would suggest to start over by making another copy of your original project file, open it and print the full list of paths with: select path from photos order by path; In your case, this is a really long list, but because it is sorted alphabetically, it should be easy to scan to determine how many different ‘root’ folders you’re dealing with. Specifically, it’s likely that some of the paths point to your Downloads folder, while others point to Desktop. You should then issue one or two update commands accordingly and print out the whole list again. Again, because it’s sorted it should be easy to scan and ensure that all paths now start with “/Volumes/LaCie Hale/Research photos”.
Finally, when you open the project in Tropy again, make sure that you’re opening the correct copy of the file. At start-up Tropy will open your most recent project at its original location. But this means that if you make a copy of the project file, then edit that copy, then open Tropy, it will open the original file, not the one you changed. To make sure, you have the correct copy open, use File -> Open from the menu and pick out the file you changed.