Two problems with this:
"nuclear" in this context refers to any weapon using either fission or fusion, both of which are nuclear events. This is well known terminology, and the term "nuclear weapon" is widely used, to refer to either kind.
As to the details of nuclear weapons (all of this from unclassified sources, which means some details, especially of thermonuclear devices, may be wrong):
Fission devices were the first developed. There were three kinds:
1) A "gun" type device, where a sphere of highly enriched uranium (HEU) has a hole in it to make it sub-critical (not reacting at a high level). An explosive charge drives another piece of HEU into the hole, completing a super-critical assembly, and it explodes. This is what was dropped on Hiroshima. It was never tested before being used in war - the scientists and engineers at Los Alamos knew it would work.
2) A plutonium implosion device. Plutonium cannot be used in a gun-type device due to the too-high natural fission rate of the material (due to contamination). So instead, a spherical shell of plutonium is coated in a spherical-ish high explosive, with many detonators around it to cause a spherical implosion shock wave. That drives compresses the plutonium to super-criticality, causing an explosion. Less plutonium is needed to reach criticality, so the bomb can be smaller.
This type of bomb was tested at Trinity Site, New Mexico, because they were not sure its would work. Originally, it was quite large. Now, fusion bombs with yields of 600 kilotons are a maybe 50cm in diameter and twice that in length.
The third type of fission bomb is like the plutonium bomb, except it uses HEU.
After fission devices proved practical, it meant that a trigger for a thermonuclear device was possible. Work on that proceeded mostly independently in the USSR and at Los Alamos, New Mexico, USA (and Sandia Base, Albuquerque, NM). Sakharov of USSR and Edward Teller (and associates) came up with different, workable designs.
Contrary to popular belief, most of the energy from a thermonuclear weapon comes from fission. It's a three stage (minimum) device: a small fission device, which provides massive heat and compression (via X-rays, the container and material in it) to the fusion portion. There, fusion takes place as described, but the fast neutrons from that fusion are used to "burn" (fission) more uranium, from which the bulk of the energy comes.
This device can be scaled up by using the fusion stage to trigger another fusion stage, etc.
There is also an intermediate device which is two stage. This is a fusion boosted fission weapon, and is probably what North Korea tested in 2017 with quite high yield (>= 150kton). In this case, one has a typical spherical fission device, with fusion material inside, resulting in fusion and additional fission energy. This sort of device requires less knowledge as the physics are pretty simple, and if you can make an implosion fission device, the fusion boosted device is a relatively simple step beyond that.
During the Cold War, a very large number of nuclear weapons were deployed. We normally think of the strategic nuclear weapons - high yield warheads carried on long range missiles to destroy cities or military installations.
But tactical weapons represented the bulk of the weapons. These were designed for use in a more conventional sense - essentially, really high yield bombs, missile warheads, torpedoes, and even depth charges (I used to ride a few feet from those during my US Navy Air days). One of the biggest potential dangers of the Ukraine situation is that Putin might use tactical nuclear weapon if his imperial designs fail.
Tactical nukes can have yields from 10 or more tons of TNT up to maybe 100 kilotons (I don't know where the dividing line lies).
