[UPDATE: SOLUTION] For this specific case, I tracked down the location of the actual system icon file in the database and removed its value, which I recognized as the value of the image I had deleted through Studio > Storage. Returning to refresh the Designer > Site Icon page, my new icon then loaded successfully. [Suggestion] I think the deletion of an image in Studio > Storage should probably trigger a search on the system icon to see if the deleted file and system icon file are one and the same. If so, it should then trigger the appropriate deletion of (1) the icon value in the database as well as (2) the appropriate transcoder image(s) in the database.

[PROBLEM] In Studio > Storage I deleted an icon image and added a different image, then resized it for Apple, Facebook, and favicon. I subsequently went to Designer > Site Icon and was not surprised to see broken image displays in Apple Facebook and Icon. I also got 404 codes for these images:

 image_transcoder.php?o=sys_icon_{apple|facebook|favicon}...etc...

When I tried to upload a new file, the Designer page gave me an error "Cannot remove current icon."

I confirmed the transcoder image (identified by its handler) was deleted from the database [edit: at least, in one table], making sense of the 404 (file not found) errors sent to the browser.

In the Designer script, I see how when one submits a new Icon through the form, if the script finds an id for the site icon but it cannot delete it from Storage (because I had already deleted it), then it simply gives the error "Cannot remove current icon," which matches the error message I received. So, what should I do at this point? 

The only thing I can think of is to change the file handle of the new file to the old file handle in the database. But in case there is a little more to it than that for an appropriate fix (like the encoded filename), I need some advice.

Please pm me as I believe I have a solution in the making here, but want to confirm some other details and, most importantly, if I am on the right track. [NEVER MIND :)]