To my knowledge our missiles and other tactical nukes and most bombs are not H bombs.
Where did you get that idea? Last I studied the topic, we had a mix of six or seven different nuclear missile warheads and gravity bombs, all of which were classified as "thermonuclear". That means the design is two-stage, with fission explosions igniting a fusion device. Although the yield is much lower than during the early days of the Cold War (and sometimes actually adjustable), these use isotopes of hydrogen. Therefore, "H-Bomb" is the colloquial term.