You are correct.It is a fusion of the atomic nucleus at very high teperatures.
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.)
Yes. Their mistake may have been using just one fission trigger, instead of multiple.
A thermonuke is none other than a hydrogen bomb
a bomb whose violent explosive power is due to the sudden release of atomic energy resulting from the fusion of light nuclei (as of hydrogen atoms) at very high temperature and pressure to form helium nuclei (webster)