The way I understand it is that it is dependent on how you label things... whatever you decide to call "0" or neutral or ground, is your reference point. And if you happen to measure something MORE NEGATIVE in potential, it is negative voltage. Like the battery examples show.
Imagine you are on a huge mountain somewhere, infinite in size. Anything below you is called NEGATIVE and any point above you is called POSITIVE. There is no top or bottom, the mountain is infinite. All that matters is the relationship between you and another point on the mountain. It is meaningless to ask you "what altitude are you on the mountain" because it is infinite. The only thing you can do is compare 2 points.
Same goes for voltages. If you call the "+" end of a single 9V battery "0" then the "-" end becomes -9V. If you call the "-" end of a single 9V battery a "0", then the "+" end becomes +9V. It is all basically what you decide to call "0". There is nothing magic about a negative voltage or positive voltage. You can even flip a voltage with your multimeter by flipping your probes around... all of a sudden the multimeter will show a negative number instead of the positive number.
If you are troubled by this, then simply call the most negative voltage in your circuit a "0" and then all the other voltages would be positive in relation to it. However, when you start working with opamps it is more convenient to think about "+" and "-" voltages. There are many instances when you want to think in terms of "+" and "-".
You may also see a central tap on a transformer.... so you have 3 wires coming out of it. If you call the middle one "0", then you have "-" and "+" voltages available between the other ends of the remaining 2 wires. That is often how you see power supply rails in systems.