These MOT based spot welders transfer far more heat to cells than capacitive discharge welders. This can result in the damage to heat sensitive components inside the cells which is the same reason that soldering isn't recommended for lithium cells.
What is the cause of the excess heat transfer? Is it mainly an excess pulse duration? I notice that the project in the OP was being pulsed on the order of 100s of ms, long enough to get a visible glow. Or is the excess heat caused by the fact that the OP runs at constant current for the entire duration of the pulse, where a capacitive discharge design is going to have the current falling off during the pulse? Or is it a blend of both?
It's excess pulse duration coupled with insufficient current. Capacitors can dump all their energy almost instantly (on the order of 1000s of amperes), but transformers are much more current-limited due to their design.
It's analogous to using a press vs a hammer.
I used to get a kick in middle school disassembling disposable cameras (unlimited supply since dad worked in a grocery store with a photo lab), charging the circuit, and discharging on coins.
To get around those issues, I built a capacitive discharge spot welder for tabbing lithium ion cells a couple years ago: http://robruark.com/projects/welder/welder.html