Ok I wasn't sure about this because the rulebook does state at the end of the runners turn. This has never come up in one of my games but I was going to play that if the runner were at say at -1 with two actions left, he would be able to use the last two actions to come back to 1 card preventing a flatline.
So what I am hearing is if the runner is ever negative even on his turns with actions to spare to get back to a positive number of cards, the runner loses.
The "end of the runners term" quote you are referring to is in reference to a hand size reduction caused by suffering brain damage.
Basically,if the runner ever has to discard a card from his hand and is unable to do so (discards down to -1 or lower), he immediately loses the game.
If the runner ever has his hand sized reduced to less than 0 for whatever reason (currently only possible via brain damage), then he will lose at the end of his turn.