Yes, cățel=pup, if you are nitpicker and you want puppy, then the right Romanian word is cățeluș. They are both common and can be used interchangeable. A small or young dog is cățel, a small AND young, or very small, or very young, dog is cățeluș. However all of pup, puppy, little dog, doggie, etc. should be accepted. If not, you should report it. A guy who tells on friends/colleagues, etc, is also cățel, in pejorative sense.
cebalanguages - As xavi75724 confirms "cachorro" seems to mean pup/puppy in Portuguese. Nice connection to the Latin! Note that there is an obvious similarity between "caine" (with the ["hat"] diacritic on the "a") in Romanian, and what, I believe, is a Brazilian Portuguese word for dog, that is, "cao" (with a tilde diacritic above the "a", thereby nasalizing that diphthong). [In other words, caine almost sounds like cao.]
In context, there is no confusion, everybody will understand it. As the word "catel" (without diacritics) does not exist in Romanian, most of the people will realize its meaning even if without diacritics and out of context. For other words which may have different meaning, it will be confuse, but it is not the case here. One example coming to mind is "fata" and "fața" ori "față", hilarious constructions may result if you omit the diacritics, like for example "I am looking inside of your girl" (instead of "looking to your face").