Except for the reference angles (which are always non-negative, acute, and measured from the nearest part of the $x$-axis), all the values in these diagrams are signed values. In particular,
- The labels for the standard position angles start at the positive $x$-axis and are positive for anticlockwise and (although there are no examples of this in the given diagrams) negative for clockwise.
- The labels for the legs running along the $x$-axis are signed lengths, positive to the right and negative to the left.
- The labels for the vertical legs are signed lengths, positive for up and negative for down.
- The labels for the hypotenuses are signed distances from the origin, positive for radially outward.
Of course, since there's no other direction radially than "outward", all hypotenuses are non-negative.
(If you go on to study polar coordinates, some sense of negative radii will be introduced. The method places a copy of the real axis through the origin along the ray of the given angle, with the non-negative half of the axis coincident with the angle's ray. In some ways, replacing "distance from origin" with "signed distance from origin" is analogous to extending trigonometry out of quadrant one by replacing "length of leg" by "signed length of leg". So in addition to all the coterminal aliases for points on the plane, all of these get duplicated to include new aliases for negative radii. For a point having coordinates $(r,\theta)$, the coterminal aliases are $(r, \theta + 2 \pi k)$, for any integer $k$, and the new aliases are $(-r, \theta + \pi + 2 \pi k)$, for any integer $k$.)
It really isn't proper to discuss trigonometry using triangles or the unit circle. One starts with geometric triangles, all lengths are positive. But since you start with right triangles, the other two angles are always acute. If you ever want to talk about an angle of measure greater than $\pi/2$, you will have to generalize somehow. (And talking about angle measures greater than $\pi/2$ is necessary. Not all angles are acute.) We introduce the trigonometric ratios while talking about geometric triangles and realize that these ratios are scale independent -- we could throw any geometric triangle on a photocopier, scale it up or down, and the ratios would be unchanged. This means we can pick some length to be constant in all our triangles and study a normalized subset of all the triangles, but still understand all of them by rescaling. In preparation for the next step, it is convenient to scale all the hypotenuses to be length one.
When we realize that every right triangle can be scaled so that its hypotenuse has length one, we take all the geometric triangles and do this to them and draw them together, with the "angle of interest" (the one we keep putting into sine, cosine, and tangent) in standard position at the origin. We get the unit quarter circle in the first quadrant. This is a direct consequence of Pythagoras's theorem/Euclidean distance. Well, we're certainly wily enough to continue that circle into quadrants two through four. When we do this, all the hypotenuses have length one; we don't get any negative hypotenuse lengths. But the legs point left and right, up and down. They are actually the coordinates of the point on the unit circle through which passes the terminal ray of the angle in standard position. If we extend our definition of "length of side adjacent to the angle" and "length of side opposite the angle" to allow positive and negative values, we get an extension of cosine and sine, respectively, from the first quadrant to the other three.
What good is this? Now our trigonometry functions tell us about all the points on a circle, not just the ones in the first quadrant. We can tell you where the attachment point of the coupling rods is all the way around the steam engine's wheel's rotation, without having to set up a new coordinate system in each quadrant. When the connection point is above the wheel's axle, the vertical coordinate, given by sine, is positive, and when below, negative. A similar observation applies for cosine. There are two basic mechanical motions -- linear and circular. Modeling linear motion with math has been in our wheelhouse since we studied parametrized linear equations in algebra. Now that we have a parametrization (by angle) for circles, we are in a position to model circular motion.
But we're not done. We need a further generalization. We started with trigonometric functions that could only accept acute angles. We generalized to trigonometric functions that could accept any angle from $0$ to $2\pi$ radians. But wheels don't roll around once and then stop. They keep going. And they go backwards. (Most likely, the idea of negative angles, large positive angles, and how each angle has infinitely many coterminal angles was explained when the term "angle" was defined.) We need to extend our trigonometric functions to handle all of these angles. How do we do it? By extending the unit circle to go around and around in both anticlockwise and clockwise directions. It's as if we have an infinite unit radius helix with axial period $2\pi$, but we only view it in parallel projection along its axis onto the plane (so what we "see" is a point doing infinitely many laps around the unit circle).
With that, we finish extending our trigonometric ratios from geometric triangles (acute angles) to an encyclopedia of values of sine and cosine (the unit circle, for angles in $[0,2\pi)$), and finally, to all possible real angles.
So it's not that you explain trigonometric functions in terms of triangles or unit circles. To initially define and then subsequently generalize, you need both. But then you have both tools in your toolbox going forward. If a particular problem is best approached with a geometric triangle, you are able to do so. If it is better approached with a unit circle, you can do that too. And in both cases, you know how the two approaches are related, so you can swap back and forth as needed.