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No, you can have a fusion booster, like in the "Sloika" [0] design, but for a Teller-Ulam design, that is a H-bomb, you use a nuclear primer to ignite a fusion reaction and by far the most energy comes from the fusion part. [1]

[0] https://www.atomicarchive.com/history/hydrogen-bomb/page-11....

[1] https://nuclearweaponarchive.org/Library/Teller.html

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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.

Half of the energy comes from the fissile "spark plug" that's inside the fusion material. Less than half of the energy comes from fusion.

It still very much is a fission weapon.

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