Assume, for example, that SHA-256 is broken, then the protocol would have to be changed but of course that does not result in a change to the currency.
Probably not the best example.
This would be a pretty big change to the currency. At minimum a lot of it could change hands against the will of the owners.
This would be an emergency hard fork that would not be a simple change.
It would destroy every current mining operation and force a complete restart there from CPU-GPU-FPGA-ASIC progression.
I know some notable speakers use this example as one of the "there are no problems we can't solve" message, but this would in fact truly wreck havok on Bitcoin.
Fundamentally though I'd agree with the broader point that not every protocol change is necessarily a change to the currency. Many would be entirely seamless to users. This just isn't one of those.
and the good news is that SHA-2 seems to be holding up pretty well with no successor clearly in sight even tho SHA-1 has been proven to be insecure. it also seems that these advancements come from the academic circles thru NIST competitions IIRC. we should have plenty of warning that some breakthrough is on the horizon with a need for SHA-3.