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High-level survey of this question is fine:
After reading https://en.wikipedia.org/wiki/Laser_diode I still can't tell if the electronics that enable a diode to lase are different from those that enable it to emit light. So, in general, is a laser diode a LED plus some sort of optical resonator or cavity?
Or are any laser diodes themselves electronically distinct from non-laser LEDs, meaning they don't look like a LED plus some extra physical structure to allow them to act as a laser?
I still can't tell if the electronics the enable a diode to lase are different from those that enable it to emit light
It's not the electronics, it's the optical cavity.
If the optical signal is fed back through the gain medium (the PN junction) such that the round trip loss is no more than the round trip gain, an "LED" will start to lase.
A laser diode's cavity can be formed by cleaved facets on the surface of the chip, Bragg reflectors patterned into the chip, or even external lenses and/or mirrors of some kind.
Generally, a device designed as a laser diode will also include a waveguide structure on the chip (and overlapping the junction) to facilitate low round trip loss, while a device designed to be an LED won't have any distinct waveguide structure, though there's also such a thing as a resonant cavity LED (RCLED).
LED: The voltage on the diode lifts the free electrons across the bandgap to a higher level. They emit light when they drop back to the lower level. Due to the rules of quantum mechanics when this happens spontaneously is random if no other measures are taken. The degrees of freedom in a LED allow for variable wavelengths (frequencies) and point in time. Thus the emitted photons are "incoherent".
LASER: The degrees of freedom for the photons are removed. The optical cavity allows only one (or very few) wavelengths (factors of the resonator length). And the previously emitted photons "passing by" stimulate the emission of the new photon. So most photons have the same phase and frequency. They are "coherent".
Even though the LED already has a very small variation of wavelength the LASER optics reduce that variation. The counter intuitive aspect of a LASER comes from quantum mechanics. You might think that a photon is emitted spontaneously and then would resonate if it has the right wavelength that fits the geometry of the resonator. But due to quantum mechanics the geometry of the LASER-(diode) makes it very unlikely for a photon to be emitted spontaneously or at another wavelength.
A diode laser is an LED in an optical cavity.
Diode lasers are kinda cool in that they "violate" a few laser rules:
The gain of semiconductors is so great that even though the radius of facets creating the cavity are really high (i.e., essentially flat), it still lases. (The laser equation predicts that infinite gain is necessary for a pair of flat surfaces to lase)!
There's a proof that at least three energy levels are needed for a pumped medium to lase, but semiconductor lasers only have two (because they are not optically pumped, but electrically pumped).
In order for a LED to be considered a "laser" LED, its design must be such that a certain amount of the light it produces must be reflected back onto itself, by optical (or electrical) means, so that newly created (via stimulation) photons are "in step" with the previous ones, thereby creating a coherent beam of photons.
Meeting the Stimulated Emission of Radiation requirement, is what makes it a laser!
