A thermonuclear weapon is an H-Bomb. This is a bomb "based" on a fission because the first reaction in a thermonuclear weapon is a fission reaction. The fission reaches sufficiently high temperature under containment to attain neutronic fusion of hydrogen isotopes (as opposed to the aneutronic fusion, which [luckily] occurs in the sun.) The energy released in the secondary fusion is used to irradiate a third stage of depleted uranium or some other non-fissile material which then becomes fissionable, goes critical, and detonates. The third stage absorbs enough energy from the fusion to allow a highly efficient fission. It is believed only around 1/3 of the energy actually comes from the fusion proper -- but none of the high yield would be possible without fusion.
Exact details are, of course, highly classified. But roughly, if you leave off the tertiary fission-fusion-fission to just get a fission-fusion device, the fast neutrons escape and you have a neutron bomb.
Interestingly, the design of a thermonuclear weapons involves very high technology and sophisticated physics and engineering design. By contrast, a nuclear weapon is in a sense a trivial exercise in physics but a very sophisticated effort in metallurgy, refining, and materials handling: once you've got enough uranium or plutonium, things take care of themselves. This is a huge oversimplification -- especially in terms of weapons efficiency -- but gives you some idea why this man's claims may be true: that India has nuclear weapons but has never really detonated an H-Bomb. H-Bomb design requires a LOT of testing. One US test was a "dud". A few Russian efforts are believed to have been in some early cases publicity stunts that probably failed (or partially failed.)
Cheers!
Numbered parts: 1.bomb casing I think Britain faked one of their tests also.
What you're describing sounds like flint lock rifle technology applied to nuclear weapons - click, clack, fiss, boom.
The North Korean test in 2006 was a fusion dud, too, I believe.