003: "In a dank cavern that's never known the sun, the patter of the rain me only companion, I dropped to one knee and said me farewells to me ill-fated mates. They sing no songs for fishermen, but the sea swallows us all the same." - Strict three hour time window. Where's the clue for it?
005: "When the skies turn grey and spirits sag, I climb high and look upon roads blazed long ago. I turn my thoughts to the hardships of my forebears till mine own worries drift away, and I find strength to carry on." - This has a strict, four hour long time window. Where is the clue for it?
013: "To know the sylph, you must be the sylph. I may lack for wings, but a short climb will put me in a similar position. Lights dancing in the dark before me, I ponder what mischief I might make--er, were I a sylph, that is." - Strict weather requirement. Where is the clue for it? (this was worse when fair did not equal clear)
015: "Even the strings of the tightest purse loosen upon seeing my famed balancing act. From vaulted gardens I leap onto the eastmost streetlamp, then bask as dour, cloudy expressions break into smiles and a shower of gil." - The intended weather clue doesn't actually clue the weather when he did this, but it's just a word/expression thrown in the text which refers to something entirely different, which is not what the log is supposed to be, lorewise. Also, the only time clue here is the streetlamp, which I would expect to be lit when this window is active, but it isn't. The time window is during the afternoon.
017: "The fog was so thick you could cut it with a blade. My foot slipped, and in an eyeblink I was sprawled out on a cliff, the ruins of a buried city before me. Sadly, the stinging pain in my arse detracted somewhat from my enjoyment of the scenery." - This has a strict, four hour long time window. Where is the clue for it?
018: "Merciful are the rains that fall, for they give succor to the thirsting land, and comfort the old wounds of one who journeyed too far from home." - This has a very strict ONE hour (three minute) window. It is the smallest time window in the first set of 20. However, nothing in this clue clues a time, period. Worse, the weather change comes very shortly before the window opens, so before I was cheating, if I saw the weather conditions be right in the zone, I'd wander by and try. It wouldn't ever work. That's because by the time I noticed the weather and got to the spot, even if it was the right third of the day, the window would have closed before I made it there. I never rushed for it after seeing the weather turn correct because none of the others had such a tiny window. The shortest other window was three times as long.
Looking forward at this spoiler, there are six more windows that are only one Eorzean hour long. A few of them clue the time, but here are a few that don't:
030: "Nothing like feeling the cool breeze on your skin after a hot bath. The view's nothin' to sneeze at, either..." - Where's the time clue for this one-hour window?
072: "In the shadow of those unholy rocks, the mightiest giant will look as a dwarf. Yet from the right vantage point, even one of humble stature can look upon them all." - It could be said that the "shadow" comment is a time clue, but it really isn't. There are shadows during plenty of times that aren't this specific one-hour window.
080: "I was an adventurer meself, aye. Till that day when, from atop a tree, I saw that ungodly thing—that twisted mass of metal and scale. Ran from the lake as fast as my legs could carry. Reckoned I'd be safer livin' the rest of my days out of a bottle." - No time clue period. I guess you could say that "day" is a clue, but I wouldn't even call the time of the one-hour window "day" at all, and the vast majority of the day doesn't work.