This is a very good question. I thought you were going to post in the main room. You should have.
I'm not sure there a steadfast rule on its use (Nevan, you're never around when we need you), but I'll venture a guess, if it's fine with you.
Either of the four examples you've posted are 100% grammatically correct in the use of the word "that" or lack of use. Inclusion of the word is more important in written form than in everyday speech, because a lot of what we say aloud is looser than what we write when writing that exact same thing. I often catch myself inserting the word "that" after I've proofread something I've written, without it, it has that "it needs something" quality to it. I believe that the rapid nature of the spoken word when compared to the careful retrospect that is sometimes offered in writing makes a big difference here too. Use of the word helps clarify or specify points.
It appears to all come down to personal preference.
(This entire answer is my opinion; I did not research any source or style guide to check first. I stand behind my response with utter confidence.)
I'm torn between accepting thanks for a job well done, or being a good little Boy Scout who wouldn't have perpetrated such a dastardly deed on an unknowing victim.