Not sure anyone is overthinking it, there is a proper tool for each terminal and a proper way to use each tool.
Although the tool in the original post is the common tool it isn’t the proper tool for insulated terminals.
That tool is designed for uninsulated terminals without the strain relief. When used on uninsulated terminals the male peg part goes on the bottom (opposite the seam).
Insulated terminals require a tool that compresses the terminal uniformly on both sides without damage to the insulation.
Terminals with the strain relief typically require a specific tool that is designed to both crimp the terminal on the wire and press the strain relief into the insulation.
Can you use that tool on insulated terminals, yes, I wouldn’t.
Alan