dsb, thank you very much for the input. I opened the valve, bled the air, primed the pump, and TC started and runs fine.
Regarding the owner's manual, in the past I had always missed a comprehensive table of contents in the front of the book. I always thumbed through the owner's manual Preventative Maintenance section and last night I missed the Fuel System page. In case anyone else has missed it, in the last two pages of the book is an alphabetical index of items you will find useful.
Here's a little more info about getting the TC started again after running out of fuel. Brian is correct, you can access the hand pump from the engine area after you raise the rear cargo bed, but you will need to remove the passenger or driver seat to open the injection pump valve.
FYI, to access the areas under the seats, I used a socket and ratchet to remove two bolts rearward. The driver's side is easy because the seat slides forward making easy access to the bolts. Remember to unattach the TICS armrest wiring harness before you swing the seat forward.
The passenger seat does not slide out of the way, so access to the bolts is more limited and there is little space to move a ratchet handle. But the bolts are not terribly tight and they come out pretty easily.
The most time consuming aspect of this job for me was locating the injection pump valve today for the first time. Perhaps it will be easy for you because you know what an injection pump looks like. For me I was relying on the owner manual illustration PM-23, which shows the general vicinity of the injection pump. I certainly could not locate that "view" on my ToolCat, from the rear or from under the passenger seat or the driver's seat. I also thought the valve might be at the end of the hose that leaves the fuel hand pump, and that is not the case. I certainly spent thirty minutes just looking for the valve. When I did locate it, the area did not look like the illustration PM-23. What tipped me off is the appearance of the end of the knob, which looks exactly like the photo.
To help you locate it (on at least the "C" series), I tilted the passenger seat forward, got on my knees on the ground and looked through the passenger seat access hole across the TC towards the rear of the driver's seat. This valve ends in a knurled knob above where a rubber hose & clamp connect at 90 degrees to the valve.
The valve opens with easy finger pressure. I pumped the hand pump several times until I felt resistance. Despite the owner's manual instructions, I was not able to see the valve to determine if fuel was coming out with no air bubbles. In fact, in my case, very little or no fuel seemed to spill out. But I then tightened the valve, closed things up and the TC started right away. Remember to plug in the TICS armrest wiring harness if you had disconnected it.
By the way I think the reason I got surprised running out of fuel this time, compared to many similar experiences when I did not run out of fuel, was that I was mowing deep grass, at full throttle, with the mower going-- and the time between a low fuel warning and game over was quite short.
Here are a couple photos. One is the vicinity of the fuel injector pump valve and then a closeup of the valve itself. Thanks again, all, for contributing to this productive knowledge exchange. You helped get me rolling again today. Now, back to work. Cheers