Hello דן, Dahn, Dani.... or (on U.S. documents) Dan, or Daniel,
As you noticed, we don't use the Hebrew alphabet in the U.S.
As for the rest...
- Your parents decided how to write your name on your U.S. certificate, not U.S. immigration officials or any other U.S. government officials
- You have decided which name(s) you want to use in English (apparently Dahn and Dani), not U.S. immigration officials or any other U.S. government officials
This issue is not a name change issue, but a pronunciation / transliteration issue.