My full name shows when I call strangers/people that don’t have my number

When I call people with my cell phone who don’t have my number saved, my full name shows up on their phone instead of just my number. I’m not comfortable with that for privacy reasons. I don’t like the idea that random people, businesses, or strangers can see my full name just because I called them. I looked this up on the internet and found that it's because the recipient's carrier dips into a database (CNAM), and to hide it I should contact my carrier to change it to something like just my initials. When I contacted them they said they can't do it, and just told me to download a 3rd party app called "Truecaller" and register my number as "unknown". Has anyone tried this and does this even work? I really don't get it, like won't there just be other 3rd party apps out there with my info, how does this even work? Appreciate any help on this.