Hebrew Name vs. Torah Name vs. Jewish Name?

I’ve been thinking about this for a while, and I couldn’t really find an answer to the question. I also wonder if this is a uniquely diaspora issue?

I went to day school and in kindergarten we had a “naming ceremony” where everyone’s Hebrew name was announced in front of all the parents. We sang some songs for the parents and there was a special oneg afterwards.

My parents gave me and my siblings names that would sound the same in English and Hebrew. Our Hebrew name matches our “birth certificate name,” so it wasn’t an issue. After the ceremony, our Hebrew name was what we would go by in Hebrew class.

However, some kids were given Yiddish names (likely named after grandparents or great-grandparents). Some of our teachers seemed to have a bias against these names, and told them their names weren’t really Hebrew names. It seems to me what they were saying was the names were not Israeli names or modern Hebrew names.

Outside day school, the only time I can think these names are used are when being called to the Torah or for a Mi Shebeirach prayer. Are there other instances when someone would use a Hebrew name? And if so, does it matter that someone is named “Chana Gittel” or “Ephrayim Velvel” ?

I also feel like this isn’t an issue in Israel, where a child’s name is just their name? There is no Hebrew/secular name.

Would it be better to characterize a Hebrew name as a person’s Torah Name or their Jewish Name (their name among Klal Yisrael)?