My surname ends in a silent H, which makes the last two letters, TH, pronounced like a hard T. I've read that Japanese tends to "make close approximations (of names) based on spelling, not pronunciation". So, depending on which is true, my name ends in either "to" or "tsu". Which one is more accurate?
No, its actually the opposite. You use katakana (normally) and get as close as you can phonetically to the original pronunciation. ツ would end your name like "gets" so you'd have an S sound at the end which is probably not what you'd want. I think hard T would probably use one of タ、ト、テ, which is ta, to, and te, respectively. No matter what, I think you'll have to end with an added vowel, which is not uncommon when trying to pronounce foreign words.
You can try an online site like here http://www.japanese-name-translation.com/ to help you get an idea. If its a common last name it may be that someone has already figured it out and there's a "normal" way to write it. If not, you can tweak it to get as close to the original pronunciation as possible.
You are partly true.
who is eminent composer in Wind Orchestra and Brass Bands musics. often translated and written as below.
Also a famous poker player
Many believe her name is pronounced like below
her sir name rarely read properly.
It's not the all the case that we notate specific name in katakana based on its actual pronunciations.