Product review (5 stars): Fixed my fridge that stopped cooling
My fridge had stopped cooling, and began to make a loud click sound approximately every 5 minutes, preceded by a few seconds of a wind up buzzing sound that ended after the click. After doing some research, I found out that the likely culprit was the start relay.

Opened up my fridge, and removed the old start relay, which had part number 8EA14C3 (3 long pins with connections, and a small unconnected metal stub where the 4th pin would have been). After some googling, I found out that a 2262181 part is just another id number for the same 8EA14C3 (and other relays with similar ids all similarly have an analagous Whirlpool number), so no point in ordering the more expensive and harder to find 8EA14C3.

Unlike other commenters, my relay didn't have a rattling sound when shook. However, if you squeeze the pins on the sides of the relay, you should be able to open up the relay. After opening, and upon removing the disc held inside, I saw two scorch marks on one side where the metal pins made contact, confirming that the relay was in fact damaged. Likewise, for those that hear rattling sounds, it's an indicator that the internal disc is shattered (fragments from mine broke off after lightly touching the scorched areas).

While waiting for the replacement part to arrive, I re-inserted the disc so that the metal pins only made contact with the unburnt (white) areas (despite the chipped edges), and I was able to startup the fridge again. However, as I don't know of any potential long term damage or risks that come with using a faulty component, I wouldn't recommend this as a long term solution. But it might come in handy to preserve some of your food until the new part arrives.