When the pitch of a plane increases (facing upwards), it has a
positive angle of attack, and when it decreases to face downwards, it
has a negative angle of attack.
Well, sort of. Strictly speaking, that sentence is only completely true if the flight path is exactly horizontal, and the wing incidence is zero. But a negative-angle-of-attack generally isn't compatible with sustained horizontal flight. Read on for more.
You do understand that pitch attitude and angle-of-attack can be two very different things, right? Pitch attitude is the angle between the longitudinal axis of the aircraft and the horizon, and angle-of-attack is the angle between the wing chord line-- or in some uses, the longitudinal axis of the aircraft-- and the direction of the flight path. Even if wing "incidence"-- the angle between the wing chord line and the longitudinal axis of the aircraft-- is zero, angle-of-attack will often be very different from pitch attitude. If you are under the impression that any time we see an aircraft in a nose-down pitch attitude, the wing must be at a negative angle-of-attack, then you need to do some re-thinking. Again read on for more.
Your sentence would be better changed to something along the lines of-- "If the direction of the flight path stays constant, then an increase in pitch attitude corresponds to an increase in angle-of-attack, and a decrease in pitch attitude corresponds to a decrease in angle-of-attack."
What I don't understand is that when the pitch is increased, wouldn't
the direction of the lift be facing backward, which would push the
You need to understand that the Lift and Drag vectors are defined orthogonal (perpendicular) to, and parallel to, the direction of the flight path, respectively.
Not the wing surface (or "chord line").
The flight path is also the direction of the (free-stream) airflow, which we call the "relative wind" -- this is the wind or airflow that the airplane "feels" in flight, regardless of the direction that the external, meteorological wind is blowing-- and that's why the vectors are defined that way.
Take a look at this related answer: Can we show through simple geometry rather than formulae or graphs that the best glide ratio occurs at the maximum ratio of Lift to Drag?. Note that the Lift vector is not tilted backwards because the aircraft is drawn nose-high, rather it is tilted forwards because the flight path is descending. If the flight path were horizontal and the wings were not banked, and the wings were at a positive-lift angle-of-attack, the Lift vector would point straight up, regardless of the pitch attitude of the aircraft.
Of course, the angle-of-attack can be decreased to the point where Lift is zero, or negative. In the latter case, if the aircraft is initially upright and the flight path is initially horizontal, the Lift vector will initially point straight down.1 Obviously, that is not compatible with continued horizontal flight-- the flight path will immediately begin to curve (bend) earthwards (downwards), and the orientation (pitch attitude) of the aircraft will immediately start changing as well!
So you can see that in these kinds of discussions, in any specific case we have to understand whether we're taking it as a given that the flight path is continuing on with no change in direction or speed, or not. If so, all the force vectors must sum to zero (meaning that the force vectors can be arranged into a closed polygon); if not, that constraint is removed.
Here's a more complicated answer that shows the same principal: Does lift equal weight in a climb?. Note that in none of the vector diagrams is the pitch attitude of the aircraft specified. And neither is the angle-of-attack of the wing. But the direction of the flight path (i.e. the climb angle) is. That's all the information we need to draw the Lift and Drag vectors in their correct orientation.2
Compare and contrast to the figure labelled "Negative AoA" in the original question. Generally speaking, this figure has two errors-- one, the direction of the Lift vector is shown as perpendicular to the wing chord line, not the airflow, and two, the Lift vector is pointing generally upward, not generally downward. To speak more precisely, for a cambered (non-symmetrical) airfoil it is possible for the angle-of-attack to be very slightly negative (just a few degrees) and for the Lift vector still to be in the "positive" direction. But not at the large negative angle shown in the figure--assuming that the figure is intended to represent horizontal flight, and thus a horizontal airflow. If the figure is not intended to make any representations about the direction of flight/airflow at all, then we have deeper problems-- this would suggest that the OP is imagining that "Angle of Attack" is synonymous with "pitch attitude".
There was one simplification made in that answer-- the Thrust vector was assumed to be parallel to the Drag vector, and therefore parallel to the flight path. In reality, the Thrust vector is not defined to be parallel to the flight path, so for any given direction of the flight path, the pitch attitude of the aircraft does affect the direction of the Thrust vector. But not the direction of the Lift and Drag vectors.