In conclusion, continue to refresh rake using TF to have it lineup for the a long-term dps gain (the rest of the encounter). Note the <9s rule was crafted simply because 6s (duration of TF) + 9s = 15s (duration of rake). From a perfect simulator's perspective, if your current rake is not TF-buffed, and you popped TF right now, then the latest you can refresh it is just under 9s, and it will have the buff.
I did not take into account the direct damage of rake, thanks for correcting that.
If I understand you correctly, you are using the TF glyph? Then what you're saying should be true.
In the case that the berserk glyph is used instead, rake and TF line up automatically and there is no need to pre-refresh rake. The only situation where you'll get the chance to refresh non-TF rake with TFed rake is when you are running around/disabled/have to do something else. In that case only the "short term" damage is considered, as the following rakes should line up with TF again.
Calculating the "short term" damage with 5ticks+direct damage, with updated shred/rake damage figures:
premade ilvl378, MotW only, non-crit damage
rake:5750 damage, 5750 per tick, 35 energy
shred:~12500, 40 energy
equivalent to 32.2 energy if spent to shred
The first tick of rake is preserved even if rake is pressed twice, thus it would be a dps gain to clip non-TF rake with TFed rake anytime.
Hopefully I did not forget something again.
According to your diagram the last tick of a refreshed rake with TF get no benefits from the TF, but that's false.
From my testings, the damage of the direct hit and the following ticks all depend on the stats at the moment you rake. Therefore I believe Robosaurus is correct.