Why does a hot cathode emit electrons if it is a cathode not an anode? I've been trying to learn about things like thermionics and vacuum tubes but I've run into something that doesn't make a whole lot of sense to me. So, assuming I'm not totally backwards in my understanding of conventional vs. electron current: an anode is where the conventional current enters and the electrons leave a device; a cathode is where the conventional current leaves and the electrons enter a device. Shouldn't it be the anode that emits electrons because they leave it?
 A: Your definition has it right inside there: "an anode is where [...] the electrons leave a device; a cathode is where [...] the electrons enter a device."  The device is the tube, so the electrons are entering the tube at the cathode pin and traveling to the cathode, being emitted from the hot cathode inside the tube, being absorbed by the anode inside the tube and traveling to the anode pin, and then leaving the tube at the anode pin.  Electrons in at the cathode connection, out at the anode connection.
A: A thought experiment might help.
Pretend that you have not heard of diode, cathode, anode, etc.  I give you a black box with two wires coming out of it.  One is labelled cathode and the other anode.  You have a variety of basic equipment e.g. DC and AC power supplies and meters.
You quickly determine that the box provides no power.  A little later you discover the curious property that it conducts current in one direction but not the other.  If you connect the positive of a DC source to the anode and the negative to the cathode then current flows.  So, conventional current is entering the black box through the anode.  Conversely, the electrons are going into the box through the cathode.
Now, I let you open the box and you see the tube / valve and you figure out that the cathode wire connects to a hot electrode in the tube and the anode wire to a cool one.  You figure out that the electrons are being emitted by the cathode.  They are leaving the cathode inside the tube but, from the black box point of view, they are entering through the cathode wire.  It is matter of point of view: are you inside or outside the tube?  You seem to be taking the inside point of view but the definition is taking the outside one.  Note that life inside the tube is not nice.  Apart from being very cramped, it is a vacuum.
Of course, you might open the box and find a semiconductor diode or a little person with a voltmeter and a switch.
A: This answer is pretty much the same as the two previous answers,
but enhanced with a diagram.
You write

an anode is where the conventional current enters and the electrons leave a device;
a cathode is where the conventional current leaves and the electrons enter a device.

These definitions are in perfect agreement with how a vacuum tube works.
Look at this diagram of a vacuum tube. The heated electrode is connected
to the negative pole of the power supply, and the cold flat electrode
is connected to the positive pole of the power supply.

(image from Circuits today - Vacuum tube diodes)
The electrons flow from the negative pole of the power supply,
enter the tube at its bottom side and reach the heated cathode.
Hence calling this lower electrode the cathode is in agreement with your definition above.
Then, inside the tube, the electrons fly from the cathode to the anode,
because they are rejected from the negative cathode and attracted to the positive anode.
Then the electrons from the anode leave the tube at its top side
and flow to the positive pole of the power supply.
Hence calling this upper electrode the anode is also in agreement with your definition.
A: Wikipedia
An anode is an electrode through which the conventional current enters into a polarized electrical device.
This contrasts with a cathode, an electrode through which conventional current leaves an electrical device.
As an example in a X-ray tube there are two electrodes connected to a dc power supply.
One of the electrodes is connected to the positive end of the power supply so the conventional current (flow of positive changes)  from the power supply enters this electrode and it is called the anode and at the other electrode conventional current leaves the electrode and flows into the negative terminal of the power supply and this electrode is called the cathode.
When you consider the flow of electrons all that needs to be done is to reverse the flow of charges.
So electrons enter the electrode connected the negative terminal of the power supply and leave the electrode connected to the positive terminal of the power supply.
In the gap between the two electrodes electrons leave the cathode and eneter the anode.
