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.