The test runs fine today. It actually stopped at the edit button click just once out of 10 or so times.
I think I figured out the problem and that was exactly my suspicion - there is another button earlier in the DOM which matches the given search criteria. It's hidden and does nothing at this point of the test but causes the problem. I'm attaching a screen capture of the DOM from the recorder where you can see it's the exact same markup. Unfortunately that means you can only use XPath or TagIndex to locate the correct button to click.
BTW, an easy way to spot such problem is running the test with Annotation on. Test Studio highlights the exact element rectangle on click so you can clearly see if it locates the correct element or not. If it doesn't highlight at the expected place this means it locates another element. In such situation just open the DOM and use the search option to find the similar (or absolutely exact) element instances.
I hope this helps!