Actually, lethal damage reducing you to 0 hp doesn't make you start bleeding out.
In the "Injury and Death" section of the Players Handbook we are dealing with a few different concepts, as well as a few different states. Let's look at the states first (Emphasis mine):
Disabled (0 Hit Points)
When your current hit points drop to exactly 0, you’re disabled.
This, in short means that you're on the verge of unconsciousness. You can barely take any actions or you'll be unconscious.
Dying (-1 to -9 Hit Points)
When your character’s current hit points drop to between -1 and -9 inclusive, he’s dying.
This is, as you explain it, basically bleeding out.
Dead (-10 Hit Points or Lower)
I suppose this one's pretty self-explanatory.
There is of course also the state of being at 1 hp or higher, which would just be alive and well.
Now, the above references your current hit points. Current hit points is your maximum amount of hit points minus any (lethal) damage you've taken.
Now, for nonlethal damage (again, emphasis mine):
Dealing Nonlethal Damage
Certain attacks deal nonlethal damage. Other effects, such as heat or being exhausted, also deal nonlethal damage. When you take nonlethal damage, keep a running total of how much you’ve accumulated. Do not deduct the nonlethal damage number from your current hit points. It is not "real" damage. Instead, when your nonlethal damage equals your current hit points, you’re staggered, and when it exceeds your current hit points, you fall unconscious. It doesn’t matter whether the nonlethal damage equals or exceeds your current hit points because the nonlethal damage has gone up or because your current hit points have gone down.
Non-lethal damage is not subtracted from your hit point total as lethal damage is. If your nonlethal damage equals your current hit point (Your total hit points minus any lethal damage taken) you fall unconscious.
Meaning, that if we have 35 hp, we take 30 lethal, we have 5 current hit points. We then take 5 nonlethal damage, our current nonlethal damage is now equal to our current hit points and we fall unconscious.
Edit: as mentioned in the comment, we of course go staggered when the total nonlethal damage reaches our current hp. We fall unconscious when our nonlethal damage taken exceeds our current hp.