I think there is a link in there to the TSA letter that states that violins must be allowed as carry-ons if there is bin space. Please note the if. That is a big if. If the bins are full by the time you board, the airline is not required to remove items to make room for your violin.
The other problem that people sometimes run into is if they are booked on a smaller commuter jet with smaller bins. In that case you may well be told that those bins will not accommodate a violin case. It has been my experience that you will never win an argument with an airline employee.
Here are some of the documents listed in the first link: https://www.transportation.gov/airconsumer/air-travel-musical-instruments
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CARRY ON AN INSTRUMENT
Guitars and smaller musical instruments like violins or flutes will be accepted as your free carry-on bag item on Delta and Delta Connection flights. Your carry-on must easily fit in the overhead bin or other approved storage locations, in the cabin, based on available space at the time of boarding. Please note, if overhead bin space is limited or if you prefer, you may check your instrument at the gate.
1. Codeshare partner baggage policies may differ. Not valid for travel to and from the United Kingdom.
https://www.delta.com/us/en/baggage/special-items/musical-instruments
FWIW, I've never had a problem even flying to and from the UK, although I am more likely to use Virgin Atlantic and American. The bigger problem is in the regional puddle-jumper, where there is usually just enough room overhead for a shaped case. Boarding early, or upgrading to premium economy, generally forestalls the worst problems in any case (heh).