I can confirm that it shows up as both for me. I’m not in US or Mexico, and it shows as “Gulf of Mexico (Gulf of America)”. The info I saw was that it shows as GoA in the US, GoM in Mexico, and both everywhere else. At least 2 of those are true, I’m guessing all 3 are

Another article might have included a photo or photos, but the ‘image’ being referred to is the image presented by a person or situation. Which is described. It’s not about one particular photo

It’s a joke, implying that ‘child’s presence’ referred to his own presence, or Trump’s.