A coil that is shorted "internally" has one or more shorts between its windings. This can happen if the insulation on the wires is somehow compromised, e.g. by heat, or by vibration causing the insulation to crack and peel off or abrade away). A coil with an "internal open" has a break in one or more if it's windings. "External shorts" probably refers to shorts external to the coil, like on the circuit board (if the coil is, indeed, installed on a circuit board. External opens are just as they sound, an open external to the coil.
When using an Ohm Meter to test for shorts, you must isolate portions of the circuit to determine where the short is (unless your ohmmeter is sensitive enough to read miliohms, in which case, you may be able to estimate where the short is, by using ohms/inch [ohms/mm] values. But, most ohmmeters aren't that sensitive, so to determine if a short is internal or external to the coil, you will have to disconnect one of it's leads from whatever it is connected to.
