Precisely; it isn't the voltage that's lethal, it's the current. If the current isn't available, the voltage is pretty harmless. High-voltage mains power lines carry a /lot/ of current, whereas high-voltage 'Tazers' don't.
OK, I'm not familiar with arc welding, and I'm surprised that 12V is enough to strike an arc. But your comment that it is the high current that matters for creating a weld, rather supports my contention that current is more dangerous than voltage.
Electron beam welding (which you probaby won't find happening in a garden shed or on a building site) requires many thousand volts, but only a small current (less than 1A).
The only welding I've ever done myself, involved a coke-burning forge and a heavy hammer.
The only 'isolation transformers' I've come across, are earthed - and either have an output restricted to 50V or less (that threshold at which the human body's resistance drops substantially) or provide a good earth connection for the appliance. An isolation transformer /does/ disconnect the connection that exists between the power station and the ground and is carried by the 'live' or 'hot' cable, but that is /not/ the same as the safety 'earth' connection which is meant to provide a relatively safe return path in case exposed metal parts of the appliance accidentally become connected to the power supply.
The 'earth' connection provided by an 'isolation transformer' to the appliances running off it, may be arranged to effectively create a 'short-circuit' on the output windings of the transformer if any current flows through it, and that should trigger the safety cut-off (probably a 'residual current device'). Or the 'earth' connection will pass straight through to the same 'earth' as the mains supply provides.
A double-insulated appliance has no exposed metal parts that can become connected to the power supply, and that's why they are permitted to be used with no 'earth'.