How can I remove 'docked' items in the Cinnamon menu ("start") bar on Linux Mint?

You just need to drag it over the application menu, though you're right - it seems like any of the ways you mentioned should do it. (And agreed, none of them do for me either).

Do the below: Right click on your Menu start Icon -> "Configure" -> "Open the Menu Editor" -> Scroll to and select the menu item you want to remove -> Delete button on the right hand side. This should remove the menu item you want to get rid of. You can also edit shortcuts via this way.

Hover mouse over desired left side menu item and press Delete, this will automatically remove the item selected (ex: Firefox, Software Manager, Terminal, Files, Shutdown, etc.).