A "neutron bomb" or enhanced radiation bomb is typically a low-yield nuclear bomb where the destructive radius of the blast is lower than the kill radius from the neutron and gamma rays. They were first developed with two roles in mind: tactical use on a battlefield and in an anti-ballistic missile role.

On the battlefield using tactical nuclear weapons against enemy forces is very much a tricky business as the very nature of warfare means that massed friendly forces will necessarily be close to massed enemy forces across a front-line. More so, advancing the front-line means that friendly forces will occupy the areas in and around where the enemy forces were. Using large city killing nuclear weapons in such situations would result in not only heavy levels of destruction and damage to friendly forces but the generation of huge amounts of radioactive fallout which would contaminate the battlefield and the entire region. To entertain the idea that use of tactical nuclear weapons within a conventional battlefield would not rapidly spiral out of control into a full scale thermonuclear exchange and eradication of population centers was, of course, a fantasy, but it was one that many in the military were willing to entertain at the time. For that reason many of the short-range tactical battlefield nuclear weapons developed in the 1960s (especially many kinds of nuclear artillery shells) were intended to be "enhanced radiation" weapons. The idea of such weapons was that they could be used to attack enemy forces without creating the enormous destruction or generation of lethal fallout that would devastate the civilian population (or friendly forces). However, to be clear, use of such weapons would still cause widespread destruction of ground structures and creation of large amounts of fallout, just less than would be produced by using larger nuclear weapons.

The other major use of enhanced radiation weapons was in an anti-ballistic missile (ABM) anti-nuclear warhead role. The early generation of ABM interceptors in the 1960s could not get close enough to the target warhead to ensure a kinetic kill or a kill with a conventional explosive so instead they used low-yield nuclear bombs. It being preferable to have a low-yield (few kiloton) very high altitude nuclear detonation over a populated area than a large (multi-megaton) detonation at an optimal altitude. Because ABM interception occurs at a high altitude where the atmosphere is thin blast effects are reduced while radiation effects are less quickly attenuated so they have a longer range. The intended mechanism of action at high altitudes was for an intense neutron flux to radiate the fissile cores of the target warhead and induce fission reactions in them, potentially enough to melt them but hopefully enough to render them inoperable (inducing a fizzle or pre-detonation like state) while also potentially damaging their electronics components.

In practice enhanced radiation weapons tended to be low-yield weapons with casing materials that were neutron transparent (like aluminum) or enhanced neutron production (like beryllium) instead of being high-Z materials or fissionable materials that could enhance yield (as is the case in a typical fission-fusion-fission compact thermonuclear warhead). Some weapons such as the W-70 are thermonuclear weapons with enhanced radiation effects but would still produce enormous physical destruction.

The idea of a high-yield warhead that produced minimal physical destructive effects while producing enough neutron radiation to have a very large lethal radius is a myth. It simply is not possible to create a high yield warhead (at hundreds of kilotons or megatons) which could kill off the population of a city while leaving the buildings intact. Neutron radiation may be somewhat penetrating but it attenuates rapidly in the atmosphere and any high yield device with a lethal radiation radius of a few kilometers would also have a similar blast radius. Such concepts were yarns spun by Soviet propaganda makers to turn public opinion against the deployment of tactical nuclear weapons in Europe. (Edit: it's more accurate to say that they were yarns seized on by Soviet propaganda, though no doubt many aspects of them were fully homegrown in the West, but the Soviets were able to use them very effectively in the court of public opinion.)

I want to complement (and compliment) the answer written by u/rocketsocks by providing a bit more detail on the tactical problems that the United States and NATO militaries thought could be solved by "neutron bombs" or enhanced radiation weapons (ERW). These problems, and thus the "use case" for ERW, don't seem to have been very well understood by (or explained to) the public at the time, which led perhaps to the caricature of ERW as a way to kill the people but leave property intact.

NATO's problem was defending West Germany from a Soviet-led invasion that might well be conventional (non-nuclear) in its opening stages and might well come by surprise. In the 1970s NATO did not believe it had sufficient strength to defeat such an invasion using conventional weapons of its own and thus might have to use nuclear weapons as force multipliers. Instead of destroying a Soviet battalion of say 30-40 tanks by firing hundreds of conventional shells and missiles from dozens of platforms, it would be much more efficient to hit that battalion with a few nuclear weapons, maybe even just one. Hence the enthusiastic deployment of small battlefield nuclear weapons since the 1950s, most infamously the Davy Crockett recoilless rifle but also shells for 155mm and 203mm howitzers and warheads for tactical artillery rockets of which the MGM-52 Lance was the newest generation coming online in the 1970s. Nukes for everybody!

But remember, NATO was preparing to defend friendly territory against an enemy invasion. And West Germany is not a vast desert: it is (particularly in the northern and central regions of NATO's expected frontline) a densely-populated modern country. NATO officers were mindful that villages or larger settlements are encountered on average every 2.5 km in West Germany. Those settlements might well have civilians in them since evacuation might have been imposible, and in any case you can only say "We had to destroy the village in order to save it" so many times before it becomes apparent you aren't actually defending anything. In other words, the West Germans and others would understandably become dissatisfied with a defensive plan that entailed the destruction of their country. So, as much as possible, you have to try to hit the invading Soviets (etc.) without also hitting the villages. Kill the people (invading military personnel) while sparing the property (West German villages and, likely, villagers).

At this point it's useful to look at the technical problems. Ordinary nuclear weapons convert much of their energy into blast and thermal effects. This is good for destroying, say, a city or airfield, but the armored vehicles you're trying to kill here are relatively well-protected. It turns out the armored vehicles don't protect so well against what is called prompt radiation, the initial dose of neutrons and gamma rays from the explosion itself that can kill or incapacitate. You could scale up the nuclear weapon to increase the radius at which prompt radiation kills the enemy soldiers in their tanks and other armored vehicles, but the problem is that you've also scaled up the radius of blast and thermal effects that are going to knock down West German villages. And in fact since the ordinary nuclear weapon delivers more blast and thermal than prompt radiation, you've scaled up what you don't want proportionally more than what you do want.

What if you could just scale up the prompt radiation? Well, in theory you can: that's the miracle of ERW. Basically, the weapon design is optimized to deliver prompt radiation by using much of the neutron flux from the fusion layer as a weapon rather than directing it into jacket of fissionable material. (Thought the OP is correct that there was still a lot of blast and thermal: reality never quite met expectations.)

There are a few other advantages. By keeping the yield low, not adding that additional fission layer, and focusing on prompt radiation, you create less fallout and other lasting effects, so you may be able to retake the territory you nuked and push the frontline back. The low yield also may seem less escalatory to the enemy. And this is a reasonable set of design tradeoffs for a weapon that, at the low end, has to fit into an artillery shell less than 6.1 inches (155mm) around.

So that's the use case for ERW in this context: making these fairly small-scale battlefield strikes on tactical units of the Warsaw Pact armies as they invade West Germany. And thus we can see that each of the points mentioned by the OP has some basis in reality.

But this does not seem to have been understood by the public which tended to associate nuclear weapons with attacks on cities and similar targets. And we should also keep in mind a lot was going on in NATO defense policy in the late 1970s-early 1980s:

1. Improvement of conventional defensive forces such as fielding of new generations of tactical aircraft and armored fighting vehicles and increases in the sizes (well not so much as it turns out) and readiness of the various NATO contingents. Also new generations of conventional missiles, smart bombs, etc.
2. A big debate over "intermediate nuclear forces" in which the Soviets were felt to have an advantage thanks to their deployment of the SS-20 missile. In response, NATO implemented the Dual-Track Decision which called for simultaneously fielding a new generation of weapons in response (Pershing II ballistic missile and Tomahawk cruise missile) and attempting to convince the Soviets to agree to arms control (the INF Treaty signed in 1987). Fielding the Pershing and "cruise missile" proved intensely controversial.
3. At the strategic level, the United States was also upgrading its arsenal with new weapons (B-1 bomber, "MX" intercontinental ballistic missile, Trident submarine-launched ballistic missile, plus some others that were canceled such as "Midgetman") while to some extent attempting to engage the Soviets in arms control.
4. Not publicly disclosed: the United States was also developing stealth aircraft.

The fact the United States was pursuing ERW designs was disclosed by a Washington Post story in June 1977 while pretty much all of the above was underway, and debate continued in earnest until about 1983. It was perhaps understandable that the public debate would be somewhat confused and that people might conflate issues. Criticisms of ERW as "people killers" are grounded somewhat in fact, but the use case was crafted around precise targeting of enemy soldiers and limiting collateral damage, which would have been to friendly territory. ERW designs are not suitable for larger yields like those used by strategic weapons, and the idea of a nuclear that would kill the people in a city while leaving the buildings intact seems to be fantasy unconnected with the actual ideas for using ERW.

As a coda to this, the United States eventually decided to develop and produce battlefield ERW but not to deploy them to Europe. Soon after the Cold War ended (1993), the United States decided unilaterally to terminate its battlefield nuclear weapons programs such as artillery shells and rocket warheads and limit nuclear weapons to strategic weapons and air-dropped bombs. What had happened besides the change in threat? Conventional precision-guided weapons--i.e., "smart bombs" as showcased in Desert Storm--were felt capable of accomplishing the missions that had been assigned to battlefield nukes more effectively and with much less collateral damage and escalatory potential.