You just don't care for the taste of blood. Any time meat is served with pink juice running out of it, there will be a slight coppery taste from the uncooked blood. Nothing technically wrong with that; but not everyone likes that flavor, any more than everyone likes something like Cilantro.
FWIW, you probably didn't realize the difference easily as a child, but if your mother was commenting on safety, the odds are that the meat you were eating at the time was pork. Starting in the 1960's the USDA warned us not to eat rare-cooked pork because of the risk of getting Trichinellosis by ingesting microscopic larvae along with the undercooked meat. There was never any such warning in regard to beef. (But I suppose some people never realized that cows don't carry it.) These days, commercially-raised pork is mostly all safe, too; the most common way to get Trich in the US today is from eating wild-caught boar or bear meat.
There were some worries about beef due to mad-Cow Disease for awhile in the 90's, but it was never a serious concern in the US food chain. While the prions that cause Creutzfeldt-Jakob Disease are unfortunately not destroyed by thorough cooking, they also are NOT present in the parts of a cow/steer that Americans normally eat. Only the animals' nerve tissue may harbor them; that is, meat from the brain or spinal column area. (There was at the time a local traditional "brain-sandwich" dish served where I live, that was made from deep-fried beef brain. Now it is only available with pork brain instead.)