Graph vs. record
Three times I've scored what looks like the best score on the graph, yet I never see that I've set a record. So is the graph inaccurate?

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It seems as though they might be fitting to a lognormal distribution rather than normal distribution?
I would love to see the raw data of solution times. Normal distribution can't be the case since it would allow for solution times less than zero. Just did a quick analysis for one puzzle where the median solution time was 194 and the z=1.5 time was 112 and the z=+1.5 time was 582 .... 1.5sigma either 82 on low side or 388 on high side. If you take log of times, then the z=1.5 log time is 4.7185 and z=+1.5 log time is 6.36647 giving a lognormal 1.5sigma estimate of 0.549 on low side and 1.099 on high side ... closer to equitable but still different enough to make me wonder what the underlying distribution is. 
You're overthinking this... :)
I did some calculations waybackwhen as I was trying to figure out how the points are awarded. I happened to do this when I was solving a lot of NEW puzzles, i.e. puzzles with very few or no scores. What I came up with is the following: 1. Each puzzle is assigned some kind of difficulty rating (this is WITHIN easy/medium/hard bracket) which then determines the initial time distribution along the curve. 2. The initial time distribution is: lowest=x, "25th percentile"=1.5x, median=2x, "75th percentile"=6x, highest=10x (e.g. 21031542012602100) 2a. Notice that "25th percentile" = (lowest+median)/2 and "75th percentile" = (median+highest)/2 3. Once times start rolling in, I am GUESSING that the only things that move are the lowest (i.e. record), the median (which, again, I'm guessing, is a true median), and the highest. 3a. "25th percentile" and "75th percentile" continue to be calculate as a simple arithmetical average of the numbers surrounding them. So as far as the distribution goes it's 50% lower than median and 50% higher than median with lowest and highest specifically noted. In essence, the graph has nothing to do with the actual distribution of the recorded times. At least that's one opinion, since we have no access to actual data. Hope this helps. 
Sometimes it can take a few minutes to register as a record. Check a few minutes later and you will find that your record will be acknowleged.

From what I can see, for the puzzles that have records, the highest score on the graph is 5 times the median. The other two numbers on the graph are just the mean of the median and the quickest record and the mean of the median and the 'slowest' score, which is technically just 3 times the median. I presume, as alamakota said above, that the median score is an actual true sample, as is the quickest score. I think the bell curve is just there as a visual aid since it never changes. Even if all of the puzzle solving times did wind up as a normal bell curve, you'd expect some variation between puzzles if you were graphically representing the distribution of solving times.

Everything that fromalama wrote is correct.
To answer the original question, I've been taking screenshots of the graph, and I put several of those all in one image, and hopefully it's useful: http://i.imgur.com/JpQ2mcR.png If you get an orange star, then you got a new record. If you don't get an orange star, you didn't. In my examples here, I got a new record in case 1 and 2, but not 3 or 4. In case 2 (a new puzzle), it doesn't look like I got a star from the graph. In fact, it shows me as higher than the median. But the graph values are completely made up, as evidenced by the circled area. So, just ignore the graph. My time was terrible, but it is the best recorded time for this relatively new puzzle, so I got the orange star. In case 3, I'm close to the left edge of the graph, but 84 is higher than 76, so I still didn't get a record. Even if I had a time of 76s, I still would have only tied the record, and I would not have received an orange star. I would have needed a time of 75s or less to get a record. In scenario 4, it looks like I have the same time as the far left of the graph, but I didn't get a record (because there's no orange star). Like case 2, this is a new puzzle, and there's not enough data to for it to show me full details, thus we can assume that the entire graph is completely made up. It's possible that someone had a time as good as 70s, but it's not going to show it, just because it doesn't have enough data. If puzzle 4 was not a new puzzle, and my time was 140s, and the best on the graph was 140s, and I didn't have an orange star, then we could assume that I would have tied the record. In that case, the chart labeled "This puzzle's statistics" would have 6 lines instead of 5, and the new line would be "Record Time", and it would show me the name of the person who I had tied. If you still have questions, please reply. 
Probably someone else already solved with same time
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For instance, I have solved the 'parking ticket' one 3 times and its median has been 689, 943, and 924. Assuming this is a mature puzzle, I would expect the best time to be essentially constant, or changing by only small amounts. So why would the media be varying so widely? 
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I would expect the median times to settle out for each puzzle, and I would be surprised to see that much variation for a single puzzle. It would not surprise me to see that much variation on a puzzle story. 
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