From the pictures and your description, I think it is likely that your diode is fine.
If the regulator isn't working, the problem is going to be somewhere else.
Diode D1 (the one you asked about) is there to protect the regulator in case you accidentally connect the 12V to the barrel jack with the polarity reversed.
The cathode of D1 goes to the 12V line, and the anode to ground. In normal conditions, it will not conduct at all.
If your power supply has the connections to the barrel jack backwards, though, D1 will conduct.
This causes a short circuit. The LM2596 only gets -0.7V on its input - it can survive that.
The short circuit also causes the green polyswitch F7 (marked p 150) to open, and disconnect the module completely from the powersupply.
If D1 is bad, you should also check F7.
D1 isn't a critical part. I couldn't find out what its markings mean, but given its job, any diode with a reverse voltage well over 12V (say, like 50V) and a forward current of a couple of amperes should do fine - if you need to replace it, and I really don't think you need to.
You measured "less than 1V in both directions" for D1.
Since you measured that with the part still on the board, it doesn't mean much. You have the whole regulator in parallel with the diode.
You need to raise one end of the diode and test it again.
I expect you will find it is OK.
Best Answer
It appears to be an 0805 case (you can confirm with a ruler). If I look at current Digikey offerings there are only Schottky 1A diodes in that package, however the package is not the same (has castellated ends).
I can, however, find offshore 1N4148 equivalent with an identical looking package so that might be a good guess.
You can desolder it and measure the Vf at some reasonable current, and thus rule out Schottky diodes. You may be able to measure the capacitance which will rule out many rectifier type diodes.
The inexpensive Chinese LCR component measuring devices show 3pF on a regular 1N4148 and Vf = 691mV. Schottky diodes will measure much less Vf and rectifiers will generally measure much more than 3pF capacitance.
Of course it could be something else like a varicap diode, if you know something about the circuit you may be able to figure it out.