Food Safety Issues/Precooked ingredients safety
I boiled four 7-8 oz. potatoes, put them on a rack, then peeled them, cut them roughly in thirds or fourths and had them in the refrigerator within 30 minutes of cooking. About two hours and forty-five minutes later, I diced them and heated them in a casserole--another 30 minutes. About half the casserole was left, and it got back in the refrigerator in a shallow dish about 30 minutes after heating. Technically the potatoes were out of the refrigerator a combined total of less than the 2-hour guideline. However, I began to wonder if the potatoes had had time to get to forty degrees (things seem to take quite awhile) before I heated them again and started the whole cooling process over, or if that made any difference anyway. After a short while I ended up putting the leftover casserole in the freezer to hurry up the second chilling of those potatoes and got them down to under 45 degrees in about an hour and fifteen mins before returning to fridge. So--is the leftover casserole safe to eat, and if so, would it have been ok if I had not hurried the chilling in the freezer?
The 2 hour consumer guideline to refrigerate foods that are capable of supporting the growth of harmful bacteria is designed to encourage consumers to not allow foods to sit at room temperature or warmer over 2 hours. It does not mean that foods must be chilled to under 45 degrees in 2 hours, but just that foods are placed in the refrigerator within 2 hours or preparation or off of the heating source.
From your description your casserole is safe to eat. In fact your meticulous adherence to trying to keep the potatoes cold throughout preparation is very admirable and shows a great desire to keep your food as safe as possible. If you were a restaurant, you would be allowed 6 hours to get your casserole chilled to 41 degrees -- 2 hours to get it to 70 degrees and another 4 hours to get it to 41 degrees. You were well within this time frame.
And yes, your casserole would have still be safe even if you had not hurried the chilling in the freezer. While this step added extra protection and safety, it's not necessary to speed chill when the entire casserole would have been at 41 degrees F or less within 6 hours.