You are correct that in a Teller-Ulam design most of the energy can come from fusion, but as a note it's generally a fission-fusion-fission design in "modern" deployments with about 50% of the energy coming from fusion.

https://en.wikipedia.org/wiki/Thermonuclear_weapon

Only very large bombs, such as the Tsar Bomba (97% from fusion) ever where primarily fusion, and those are not in modern arsenals.