Hi,George I know that Vetereinary services dont want to ignore any animal groups,it usually results in a very cluttered and busy design which is definitely not a plus when we talk about logos.
Here is my solution to the problem allowing specific attention and diversity at the same time.
These are not 2 different logos but the same logo with a different theme If I bring my cat to you, you give me a business card with a cat themed logo and of cource dog owners get a dog themed one..
The design can be developed further to include a bird,reptile etc. I have done cat and dog because these are the most frequent patients a veterinary clinic usually has.
If you like the idea I can improve the design. Best regards Stickie