How do LIGO and Virgo know that a gravitational wave has its origin in a neutron star or a black hole?

The most obvious — though possibly least convincing — way is by noting the "mass gap": the heaviest neutron stars we know of (by other means) are lighter than 3 solar masses, while the lightest black holes we know of (by other means) are heavier than 5 solar masses. So if the constituents of a binary that LIGO detects have masses in one group or the other, LIGO/Virgo folks sort of expect that the objects are really in that group. If you look at the current confirmed detections (shown in the image below), you'll notice that there is indeed a significant gap between the masses of the neutron stars and the masses of the black holes. But part of LIGO/Virgo's job is to look for things that we can't find by other means, which might show us that there are lighter black holes (BHs) or heavier neutron stars (NSs) than we expect otherwise. So they don't stop there.

It's also possible to look for "tidal effects". Before two NSs (or one NS and one BH) actually touch, the matter in the neutron star will get distorted in ways that a black hole can't. The build up of this distortion takes energy, which comes out of the orbital energy of the binary, and that loss of energy imprints itself on the orbital motion — most prominently, on the "phase" of the binary which is the most accurately detected aspect of the inspiral. So when the OP says the BH and NS "gravitational fields, except for their intensity, are identical in that space beyond the radius that defines them", that's not quite true. It's true for isolated nonspinning objects (thanks to Birkhoff's theorem, which I guess is what the OP was thinking of), but it's not true for objects in binaries, and not once you get below the radius of the NS.

That brings up another important difference: NSs merge (basically) when their matter comes into contact with the other member of the binary, which is significantly earlier than BHs come into contact with each other. The BH radius is much smaller than the NS radius, so essentially a pair of BHs get to keep going for a while, going faster and faster than they would if a NS were present. This talk of distances is a bit imprecise, so it's better to talk about the GW signal observed at large distances from the binary (e.g., on Earth). You could — in principle — see this effect in the GW signal where the BH signal would just keep getting faster and stronger after the NS signal "shuts off". Of course, it's not really shut off; complicated stuff happens after NSs merge.

After the objects merge, they continue to exhibit huge differences. For example, if there's a NS involved, some matter can get flung out in a "tail" or into a disc around the central remnant. This extra motion of the matter (that wouldn't happen if there were only BHs) can generate its own gravitational waves, which could possibly be detected directly. More likely, the NS will "smear out" and just not be as good at emitting gravitational waves, so the peak amplitude will be smaller. However, after BHs merge, we know that they "ringdown" exponentially quickly. Basically, BHs have a very fast, simple, and well understood ringdown phase, whereas NSs have a messy and non-exponential aftermath. For example, we frequently talk about "mountains" on NSs afterward, which will continue to spin and give off sort of mildly damped but mostly continuous waves. Of course, it is possible that a merger with one or two NSs will end up with a single BH at the end, which will also ringdown, but before or in addition to that, we expect a lot of other complicated stuff to happen. [Note that the binary NS merger shown in the figure below ends up in a question mark, meaning that we're not entirely sure whether the remnant is a huge NS or a tiny BH.]

I should explain that these merger and post-merger effects happen at pretty high frequencies (because NSs are relatively low-mass objects), whereas LIGO and Virgo start to become much less sensitive as you go to higher frequencies (because at high frequencies there just aren't enough photons arriving at the interferometer's output; the number of photons per period, say, becomes quite random and therefore noisy). So it's not entirely clear whether or not we'll be able to see the "shutoff" or "mountains" with current detectors. A lot depends on unknown physics, and our ability to create good models for the signals given off by merging NSs. But it is true that we have not yet seen any direct evidence for them as of early 2019. So the last two items I described have not yet featured in claims about whether any source involves NSs or BHs.

But one thing that will tell us for sure if there was much matter involved — and was the reason we were so sure about the binary NS LIGO/Virgo announced in 2017 — is the presence of electromagnetic signals. Obviously, a pair of BHs on their own won't give off any obvious electromagnetic signal, whereas those huge amounts of matter when a NS is involved should give off some signal. If we detect an electromagnetic "counterpart", we can be much more confident that there was a lot of matter involved; if we don't detect any, it's unlikely that there was much matter in the system.

So there's no one piece of evidence that proves beyond doubt that there were only NSs or only BHs involved, but a collection of evidence that points in that direction. And really, how sure we are of the conclusion depends on a lot of factors. If the signal is very "loud" and clear, and the masses are very far from the mass gap, we can be particularly sure about our conclusions. But if the signal is from a source that's very far away, or is otherwise hard to measure, and if the masses are close to that mass gap, then we wouldn't be too sure about our conclusions. For all the systems confirmed so far, I think it's fair to say that most GW astronomers are extremely confident in the conclusions, but there are certainly more detections on the way that will be much more uncertain.

A measurement of the frequency of the gravitational waves from a binary system tells you the orbital period of the binary. The rate of change of that frequency tells you how fast that orbital period is changing. The combination of the two uniquely determines the "chirp mass". Basically, a lower mass binary system exhibits a much slower rate of increase in frequency during the chirp.

To first order, the following equation can be used: $$\frac{df}{dt} = \left(\frac{96}{5}\right)\left(\frac{GM_c}{c^3}\right)^{5/3}\pi^{8/3} f^{11/3}\, ,$$ where $f$ is the observed GW frequency and $M_c$ is the chirp mass, defined by $$ M_c = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}\, ,$$ where $m_1$ and $m_2$ are the component masses.

Whilst the individual component masses are not determined, the chirp mass does give a lower limit to the total mass $M_T \geq 2.3 M_c$ and a lower limit to the mass of the most massive component $m_1 \geq 1.15M_c$.

So for example, the first detected gravitational wave source had a chirp mass of about 28 solar masses suggesting a minimum primary mass of about 32 solar masses.

Such large masses are totally incompatible with neutron star sources. Neutron stars have a maximum mass of about 3 solar masses (and the most massive yet observed are about 3 solar masses).

In addition we can observe what the frequency at the maximum gravitational wave amplitude. This frequency basically increases until the merger of the two begins. For the first GW source that is at about 150 Hz. You can then use Kepler's third law (with some small relativistic corrections) to estimate that the separation of the two objects In the case of the first GW source this yields a separation of only $\sim 300$ km.

The combination of these two analyses (cramming $60+$ solar masses of material into what amounts to about twice the Schwarzschild radius of a black hole of that mass) strongly suggests merging black holes.

On the contrary, the merger of lower mass neutron stars yields much lower chirp masses and a much higher peak frequency and separation. However, to discriminate between the merger of a neutron star and a black hole of the same total mass requires a careful analysis of the signal just prior to and after the merger and comparison with the predictions of General Relativity. Neutron stars have a surface and are affected by tidal forces which feeds into subtle differences in the GW signal.

Finally, a clinching piece of evidence is that the merger of neutron stars is expected to result in a significant "electromagnetic counterpart" - a kilonnova - that can be observed by other telescopes at Gamma ray through to optical and infrared wavelengths. Such a counterpart has been seen in the only confirmed neutron star merger signal, but is not expected to be present when black holes merge.

There are a couple things about the gravitational wave signal from a kilonova that distinguish it from the gravitational wave signal from merging binary black holes, some of which are discussed in the discovery paper for GW170817:

• The signal provides information on the chirp mass of the system, a quantity that can be used to infer the actual masses of the components. In the case of merging neutron stars, these masses (a few solar masses in total) should of course be substantially lower than the masses of merging stellar-mass black holes (a few tens of solar masses).
• The internal structure of the neutron stars should lead to tidal deformations that are visible in reconstructions of the signal. Black holes do not exhibit the same deformations.
• There should be electromagnetic counterparts to the gravitational wave that would not be expected in a binary black hole merger, and these transients were observed across the electromagnetic spectrum after GW170817 was detected.