Primarily not guiding your piece properly. A splitter can help avoid catastrophy, but the cause was in the way you passed the piece through. It's better to know why it happend than to credit something that covered for bad technique. I'm curious though Gary - did you have adequate support under the whole workpiece, or most of it? What were you using to guide the stock through? Miter, or sliding table? What amount of the stock was supported?
Ha! See - good reason not to use those "gadgets"... (just joking for those who are funny impared)
Not true. Your table saw is perfectly capable of doing cut offs. You will see fly-by's when you allow the stock to twist as you complete the cut. If that same twist had happened earlier you'd have experienced a bind. Your miter saw might well have been a better choice, but that does not make a tablesaw the wrong tool to use for this type of cut. Again - it goes back to technique. If you use the fence for a narrow piece you will lack control - bad technique. If you use the miter on that same table saw to do this you have all the stability you need.
Not understanding what really happens in an accident and then giving credit to the wrong thing for preventing accidents from recurring in the future only guarntees another accident down the road. I made a joke above, but the point is very real. You cannot put faith in things like splitters and guards. As you saw Gary, you had a splitter in place and you had an accident. It isn't about splitters and guards. It's about what makes wood go airborn. It's always bad technique. Splitters can indeed cover for the bad technique a good deal of the time, but the bad technique is still there. That's what requires the focus with power tools. Sooner or later bad technique is going to catch up with you faster than not having idot proof devices will. BTW - I really do believe splitters are a good idea - really. Honestly.