is catelus even smaller puppy - is that a meaning?
Oh, you're thinking about "câine", "cățel", and "cățelus". Yes, the latter two refer to puppies. They come in sizes, you know! :-)
Then the english should be "my daughter wants a little puppy". And why doreste, not vrea?
Size is relative, compare a St. Bernard little puppy with a Chihuahua big pup :)
"A dori" and "a vrea" have almost identical meaning, but "doresc" is quite emphatic, in common speaking you will hear mostly "vreau".
The audio is missing, at least for me on cățeluș.