Another word for dog is canis. That's an odd choice of words.
To me, a dog is a domesticated mammal, and that's all.
But then, to each his own.
As a humanist I'd go with dog.
I don't agree with what someone else said - in my opinion a dog is an animal (anyone else? - a humanist would argue that an animal is not a person).
A dog is a dog.
A wolf is a wolf.
A wolf is not a dog.
A wolf is a member of the canid family of dogs, and is not a member of the canine family of dogs.
A dog is a member of the canid family of dogs, and is not a member of the canine family of dogs.
A dog is a domesticated mammal (anyone else?), and is a member of the canine family of mammals.
The best word to use is, as someone else said, canis.
If someone asks, "what's a dog?", you can say:
"It's a member of the canine family of mammals."
"Dachshund" is not a synonym for "dog" as a dog is a dog, and a dachshund is a member of the dog family.
I think you're correct that in the US and UK, "dog" and "canine" are considered synonyms.
The Oxford English Dictionary, however, has canine as a synonym for dog and defines the latter as "A mammal belonging to the family Canidae, especially one of the smaller breeds with a soft coat and short legs (such as the Labrador, the German Shepherd Dog, the Chow Chow, the Dalmatian, the Jack Russell Terrier, and the Poodle). Also used in the plural for 'dogs'."
In the UK, "canine" is often used, while "dog" is more often used in the US.
A person who is interested in the animal/pet world would be more likely to use "dog" to talk about a canid (especially one of a particular breed) and "canine" to talk about all dogs.
You could also use "canid" to refer to a canid in general, and "canine" to refer to any dog, but this may cause confusion as some people might think you were talking about an animal in the dog family when you actually were talking about a member of the wolf family.
However, in my experience it doesn't cause confusion in conversation, as the word "