The difference you have noticed most likely is due to the differing interpretations of the story by the translators. I have seen the title itself rendered into English as "The Lady with the Pet Dog," "The Lady with the Dog," "The Lady with the Little Dog," and "The Lady with the Lapdog." I do not believe that the dog symbolizes anything in particular. It was created to give the female character a reason for taking so many walks along the promenade and also to give Gurov a convenient way to strike up an acquaintance. He makes the little dog bark at him. Anna Sergeyevna apologizes for her dog's behavior. This allows them to enter into a conversation.
The story opens with an admirable sentence, which has been translated in many different ways. My recollection is that it is something like this:
They were saying that a new visitor had appeared on the promenade, a lady with a pet dog.
This makes it easy to identify Anna as a newcomer when Gurov sees her with the little dog. It also illustrates how boring the resort is, and it suggests how bored Gurov himself must be, if everybody makes conversation out of the fact that a new visitor has arrived and that she has a pet dog. Furthermore, the sentence suggests that the setting is a summer resort with a promenade along a body of water, and it would probably suggest Yalta to Russian readers. If Gurov had not been feeling so bored at Yalta, he would not have tried to strike up an acquaintance with this woman with the intention of seducing her just for the sake of having something to do and a means of passing the time. The moral of the story has to do with the serious consequences of frivolous behavior. Gurov not only brings suffering upon himself, but he makes the poor woman feel wicked, frustrated, and confused.
Perhaps it might be said that the tiny dog symbolizes the incident itself. This toy dog brings two humans together and changes their lives irrevocably. When it starts barking at Gurov, it might be imagined that the dog is giving him a warning to stay away from Anna. At the same time, it could be asserted that the dog symbolizes its owner's loneliness and consequently her accessibility and vulnerability. Women often keep little dogs for companionship. The dog seems to drop out of the picture completely after it has served its purpose in bringing the man and woman together.