Am I right to say an ESR meter only checks capacitance?
No, an ESR meter doesn't measure capacitance at all. It measures Effective Series Resistance (ESR) of a capacitor, without checking its capacitance value. A real-world capacitor behaves like a perfect capacitor with some resistance in series (plus with some resistance in parallel, and some inductance, but we ignore that for now).
An ESR meter measures low resistances, using a high frequency, low voltage AC signal. At the high frequency, the ideal capacitor ought to be very nearly a short circuit. But the real capacitor will have some effective resistance in series, and that's what the ESR meter measures.
When an electrolytic capacitor dries up, its effective series resistance goes up. As this resistance goes up, if the capacitor is subject to ripple, that resistance causes heating, and the heating causes the electrolyte to dry up faster. This usually results in a "death spiral", where the ESR fairly quickly goes from low to very high.
A service technician's ESR meter isn't a very high precision instrument. It's basically a binary tool that's intended to help you separate bad capacitors from good ones. That's exactly what a repair technician needs to know. Another aspect of it is that it uses a low voltage signal, low enough to avoid turning on semiconductor junctions. That way, it can be used to test capacitors in circuit, usually. It will be fooled by multiple capacitors in parallel, or by a capacitor in parallel with a low value resistor (capacitors are sometimes found in parallel with each other, but capacitors are rarely found in parallel with low value resistors in practice). But as long as the capacitor is surrounded by semiconductors and/or resistors that are much higher value than the expected ESR, the meter can be used without desoldering the capacitors from their circuits.
A higher end LCR meter can check for both capacitance and inductance?
The L stands for inductance (don't ask me why), C for capacitance, and R for resistance. An LCR meter measures inductance, capacitance, and resistance.