I've done this three times on titanium pots and mugs. I used a dremel tool, and tried several different bits. It is very easy to accidentally make a hole by cutting/grinding too fast or by prying (this happened on one of my attempts).
I would recommend avoiding cutting wheels and prying. Even the most gentle prying deforms the metal, so you will never get a clean result this way. I found that the slowest method, a grinding ball dremel bit, gave the cleanest result and preserved the thickness of the wall of the pot/mug.
I traced the tiny circular periphery of the weld spot with the grinding ball over and over again until a moat was formed around it. I periodically gave the handle a gentle pull to make sure I hadn't ground into the wall of the mug/pot. When the metal around the weld spot became as thin as foil, I moved on to the next one. When all of them were done, I jiggled the handle free and ground down the weld spot nubs until they were flush.
If you grind directly on the weld spot instead of around it, you're likely to create a cavity in the surface of the wall of the mug/pot due to the curvature of the ball bit. Below is the kind of bit I used.
It took me several hours, and I had to take periodic breaks to avoid getting impatient and going too fast.