You actually have your titles a little bit mixed up! (But the vast majority of the public doesn't even know that there IS a difference between the titles, so kudos to you for that!)

Service dogs are specifically trained to assist one person (their handler) with a disability. This can be any disability - from blindness to PTSD, as you mentioned. If they're trained to perform a task that mitigates a disability, they are a service dog.

Therapy dogs are dogs that are trained to remain calm and comforting in a variety of situations. Their job is to visit places like hospitals, nursing homes, etc. They visit places of high-stress and get lots of petting and love in order to reduce the stress of the people there. However, since they aren't a service dog, they do not have public access rights - they must be certified through a private therapy dog organization and invited into whatever building they are entering.

Emotional support animal are untrained pets that provide comfort to their psychiatrically disabled handler from their home. An example would be a dog for a victim of CPTSD, making them feel more secure after experiencing something traumatic in their own home. ESAs ONLY have special permissions in apartments and air travel. (If you see a dog labelled as an ESA in your local Trader Joe's, their owner is taking advantage of public misinformation - and breaking federal law).

And I'm guessing that this service dog was "released" momentarily to say hi to Donald. Most SDs have a cue word that lets them know that they are "off-duty" for a moment. And their handler knows (based on the nature of their disability) whether and when it is safe to have their dog greet someone. :)