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I think both you and the parent poster make valid points. The key thing is to send something vague-enough in the first email. For example, don't say "I think that's a problem with X and I'm looking into it", just say "I'm looking at it". In other words, only communicate the lock you're acquiring on the task and save the rest for the "done" email.

Sure. It's all about judgement as to whether setting that lock adds too much overhead to be efficient. If it's a small enough task, it doesn't matter if multiple people do it.

This is why bugtrackers have separate "Accepted" and "Fixed" states. :-)