I love those pictures! And I want to point out a flaw in your thinking: you were not "rude" at all, you were conveying to this woman to get the heck away from you in a very polite manner. Much more polite than many here would be, lol. Alvin is a puppy and puppies do these kinds of things. He might have wanted to play? But you did everything right!
This! And yes you did everything correctly! Your job is to protect your job and that's what you did. Your not going to get any flack here for doing your job!
Alvin let you know that an unbalanced out of control dog was approaching and you handled it!
Job well done.