My best guess is what I would call a 'Sunken Die' error, sometimes other terms like 'Die Subsidence error' are used.

You can read Dick Johnson's definition of Sunken Die here:https://nnp.wustl.edu/library/dictionarydetail/516860

In short, a sunken die has an area in the striking face of the die that becomes depressed (sunken, not sad) at some point during use. A slight depression may be difficult to detect by a casual observer of the object. More severe examples include die cracks and it is plainly evident something went wrong in manufacturing.

A few websites that discuss and show examples of die errors can be found here:

Sunken die errors were more common on objects struck prior to the Industrial Revolution which brought better quality tool steel used for stamping dies. When a die did fail, companies would commonly continue to use them with the error instead of making a new die due to the time and costs involved in making a new die. In the case of events like a presidential inauguration the manufacture (Tiffany in this case?) was likely working on short lead times which is another reason they didn't make a new die.

Below is an example of what a sunken die looks like. This die was made by Rudolph Laubenheimer and was used to make a Civil War Token:

In your note you stated, "whether it was a poor minting job, over melted in some way I don't know." Stamping dies (as were used for your medals) are used by forcing solid metal into the design in a die, usually via a press. Not to be confused with a casting process in which molten metal is poured into a mold. Your medals were die struck, not cast.

To read the earlier E-Sylum article, see:
Vocabulary Term Sought : Vocabulary Term Sought(https://www.coinbooks.org/v27/esylum_v27n24a07.html)