I understand why it's the red crystal symbol, but I still can't figure out why they weren't allowing a Star of David if they were allowing the other two.
Oh, I see.
Historical reasons play a large part. Basically, when the "distinctive signs" were being codified in various Geneva Conventions (the Red Cross in the Convention of 1906, the Red Crescent and Red Lion* in those of 1929), Israel was not yet in existence, so there wasn't any national delegation pushing to have a red Shield of David acknowledged as a "distinctive sign."
You can really see the historical progression in this. the
1906 Convention says:
Art. 18. Out of respect to Switzerland the heraldic emblem of the red cross on a white ground, formed by the reversal of the federal colors, is continued as the emblem and distinctive sign of the sanitary service of armies.
The
1929 Convention expands on this, stating:
Art. 19. As a compliment to Switzerland, the heraldic emblem of the red cross on a white ground, formed by reversing the Federal colours, is retained as the emblem and distinctive sign of the medical service of armed forces.
Nevertheless, in the case of countries which already use, in place of the red cross, the red crescent or the red lion and sun on a white ground as a distinctive sign, these emblems are also recognized by the terms of the present Convention.
Italics mine. Article 38 of the
1949 Convention says the same thing. The diplomatic conference which resulted in the 1949 Conventions started on 21-Apr-1949; Israel wasn't admitted to the UN until 11-May-1949, so it's entirely possible Israel wasn't even represented at that conference either.
Moreover, it's important to note that the recognition extended to the Red Crescent and Red Lion in 1929 was a codification of
a pre-existing practice. It's not that, say, the Turkish delegation at the 1929 conference said "can we use a red crescent instead of the red cross we're using now?" And it wasn't so much about achieving a consensus on which symbol(s) to use, as it was about agreeing not to shoot at each others' medical personnel, vehicles and facilities. Thus, the focus was on agreeing that certain "distinctive signs" denoted medical stuff and shouldn't be fired upon, and not so much on the precise nature of those signs.
In principle, Israel
should be able to demand recognition of the red Shield of David as a distinctive sign on the basis of this historic precedent, but diplomatic conferences on the Geneva Conventions are few and far between (the 1977 conference resulted in protocols additional to the 1949 conventions, not in new conventions), and I imagine the various Arab nations which were hostile to Israel for much of the period since 1948 were never willing to acknowledge the red Shield of David as a distinctive sign anyway.
* - Essentially, the Persian emblem of a lion holding a sword in one forepaw, with a sun overhead, only done in red. It was used as a "distinctive sign" only by Persia/Iran until the Islamic revolution of 1979.