This seemed to be deceptively simple, which suggests that I probably missed some important detail or misunderstood the rules. I couldn't quite follow the example on how to count moves, but the following scenario uses very few moves. I didn't read any other answers so as not to influence my solution. My solution assumes that the train can move to the right some distance on the unlabeled track, which I call "Tr0" for the mainline. The movements:
1. Before any train movements occur:
A. Add a bridge from the loading dock to MT car 5 (Tr2:C5) and add a bridge from car 5 (Tr2:C5) to Hold car, car 9 (Tr1:C9).
B. Add a bridge from the loading dock to MT car 6 (Tr1:C6).
C. Load the three cars with cotton bales. Bales will pass through car 5 to get to car 9.
Result: This step takes advantage of the fact that these three cars are lined up properly from the start. Now there are only four cars that need to be loaded and they are conveniently attached to the train on what I call Track 0 (Tr0).
2. First train movement: On Tr0/mainline, disconnect the way car from the train (Tr0:C10).
The consist, reading left to right, is C10, 11, 12, 13, plus the Engine. The way car remains in its original starting position on Tr0.
3. Move the train straight ahead on the mainline until car C10 clears the switch and then back down to Tr2.
4. On Tr2, connect car C10 (the last one in the consist) with car Tr2:C6. The consist is now, left-to-right: C1, 2, 3, 4, 5, 6, 10, 11, 12, 13, E.
5. Move the train forward until you clear the bottom most switch on Tr0, then back the train onto Tr0/mainline.
6. On Tr0, couple car 1 (the last car) to the way car.
7. Decouple car 6 (Tr0:C6) from car 10 (Tr0:C10). This leaves the caboose and cars 1-6 on Tr0/mainline.
These seven cars will fit because six are 40', and the way car is less than 40'. So, 6x40' is 240' plus ~30'. We also know that Tr0 can hold the way car and engine and two 50' cars as per our starting point. So we have plenty of room to spare.
The train now has the MT cars Tr0:C10-13 in its consist.
8. Move the train straight ahead on Tr0/mainline until MT car C10 clears the switch and then back it down to Tr2 again.
9. Uncouple car 13 from the engine. If MT cars Tr2:10, 11, 12, 13 need to be spotted "just so" near the dock for their bridge placements and loading, that will add additional uncoupling steps. If not, the dock workers can just add the bridges where ever and load up these four empties while they remain coupled.
Solved 1: The four empties are spotted.
10. The engine returns to Tr0 the mainline (2 moves?) and connects to car Tr0:C6. The consist is now: way car, C1, 2, 3, 4, 5, and 6 -- that is, it has six cars.
Solved 2: The train leaves the mainline Tr0 with six filled cars (and the way car).
Tomlinson Run Railroad
UPDATE: My assumption was that all cars in the starting position are actually coupled to one another. That now seems false but the basic steps remain. You just need to connect the cars but the order remains the same as I described it. In other words, by filling up three empties first, I've removed the need to line up cars for passing bales via bridges from a car on one track to a car on another track. TRRR