You may need to lift one lead to test so might be easier to just replace the diodes

https://www.fluke.com/en-us/learn/blog/digital-multimeters/how-to-test-diodes

Testing-a-Diode-Using-Multimeter-Diode-Mode (resized).jpg

This might sound like an odd way to test it:
Take the diode off one of the coils. Connect it in series with one of your unlabeled diode coils to the feature lamp voltage in the game.
When your unlabeled diode coil is wired in the correct direction, each coil will measure half voltage (about 3 volts each).
When your unlabeled diode coil is wired in the wrong direction, it will read 0.7 - 1.0 volts with the other coil having 5 volts or so.

BTW the coil wire in your picture is up against the metal frame. Bend the coil wire away from it so the plating doesn't wear off from vibration and short to the frame.

When in doubt, replace the diode, since you already have the coil reconditioned. They are about 0.20 a piece. 1N4004 - 1N4007 are all OK.

At the same time, remove all excess solder and remains of old wiring from the coil lug, to make it look nice and new.

Your measurements show that the diode is open (probably burnt out because if the alkaline battery gives quite much short circuit current).

When you connect the battery across diode so that banded side is negative, the diode acts as (almost) direct short and in theory you should see only 0.6-0.7V across it - before the diode burns out. You should use a series resistor at the battery, of about 100 ohms to prevent diode burning.

I'm not sure you've tested it right - none of the readings show a diode conducting in the 0.6V - 0.7V range as Tuukka mentioned.

The below images show what I meant using the same coil model:

Coil_Diode_Test_FB.png
Coil_Diode_Test_RB.png

Yes, using another coil as series current limiter is OK. But a resistor or even a #44 light bulb would be just fine.

Depending on the resistor, you would expect to see 0.6-0.7V in other direction, and considerably more on the other.

I tried a #44 light bulb in series with a 26-1200 coil before drawing those diagrams above - actually I could see a minor difference in light intensity depending if the coil diode is forward or reverse biased.