The method of operation on most of these machines is a heater mounted on a block to provide thermal mass and through which a small bore metal vaporisation tube passes, attached to the block there is usually a bimetallic thermal switch (normally closed) this is what is shown in the first picture, there is also a thermal fuse (the small axial component that looks a bit like a diode that you have also identified).
All 3 of these parts are wired in series usually across the mains with either a relay or the mains switch to turn the circuit on.
So after turning it on you can usually smell that it is warming up or feel some heat coming off if you hold your hand about 6 inches above the heated block.
If this is happening then it suggests the heater and thermal fuse are OK.
If not unplug and use the resistance measurement across the thermal fuse (up to 10ohms is usually OK), bimetal switch (about 1-2 ohms) and the heater (6-250 ohms depending on the power rating for a small DJ type unit I would expect 100-200 ohms)
Now if the unit is heating up this is where it gets a bit trickier using the meter to measure AC volts monitor the voltage across the bimetal switch, which should initially be around 0V, but should start changing between 0V and the 230V or 115V depending on your local mains voltage, this switching temperature is usually reached in 3 to 5 mins, but if this switching does not start after 7 to 10 mins it may be that the bimetal switch has failed (they sometimes weld shut) in which case keeping the unit running will probably burn out the thermal fuse (this is there to protect the heater from damage and not the fluid which is pretty inert and contrary to an earlier post is almost impossible to ignite).
If you need to replace anything in this part of the machine it is better to crimp rather than solder, the best method I have found for fitting replacement thermal fuses is to crimp them in using bootlace ferrules.