That's not the only distinction that matters though. Investigators are assigned to specific crimes. Their job is gathering evidence, not enforcing laws.
The point to needing to identify police is that without that rule and the accountability that comes with it the "police" become just another gang on the streets. None of that is relevant to investigatory law enforcement.
Obviously both kinds of anonymity can be abused, but only one has an easy and obvious solution.
Seriously: no one sane thinks that riot cops should be operating in street clothes and refusing to identify themselves. This whole subthread is a ridiculous digression.