It is only an inclined flat plane as an approximation for the sole purpose of calculating slope.
Correct. Which is the only context in which "slope" is defined. You're basically saying that "slope only means what it means when you calculate it to mean what it means." Well, sure. If you calculated to mean something else, it wouldn't be "slope."
The slope at a point on a curve is the tangent to that point.
The slope at a point on a curved surface is the inclination of the flat plane tangent to that point.
In both cases, you use the classic calculus effect of a limit of the series where shrinking the calculation of points from far apart to infinitely close generates a progression from secant to tangent.
You're basically saying "I'm looking at the limit of more microscopic considerations where, in effect, what looks like a curved surface at bigger scales now looks like a flat, inclined plane."
My statement is true: "slope" is an artificial notion, the tangent, that is an approximation to reality at some scale. What is the "slope" of a hyperbolic surface? The surface doesn't have an overall "slope." There is only "slope" at various local spots on that hyperbolic surface, where some approximation equivalent to a tangent plane can be computed.
For any location on a surface, you can imagine a vertical, flat plane, perfectly aligned in the vertical direction with the Normal (z direction) of the coordinate system. As you rotate that plane about the Z vertical axis that passes through the point, you see how where that vertical plane cuts the surface you get a line. Compute the tangent to that line and you get a slope for each such possible line. As you rotate that plane, beginning with the rotation where it gives the maximum slope for the line of intersection, you'll see the slope go down in value until reaches 0.
The proof, by the way, that at every location for which you can compute a non-zero "maximum" slope the "minimum" slope is zero can be visualized without any math, as a simple thought experiment.
Consider a surface covered by a grid of discrete point locations. For each point location, a slope value has been computed. If the surface were a flat XY plane, that is, perpendicular to the Z axis of the Euclidean coordinate system, the slope at all locations would be zero. If the surface has undulations, hills, valleys and saddle points, the slope at various locations might be zero or non-zero.
Suppose one of the undulations of the surface is a hill shaped like a camel's hump. At one of the locations on the side of that hill the slope is 45 (we'll use degrees for conceptual ease) in whatever you consider to be the "straight downhill" direction. Super. Let's teleport ourselves onto the hill so we are standing at that spot looking straight downhill, where we measure a 45 degree angle that is the slope. You've been miniaturized so where you stand looks like a flat, inclined plane (all parts of that surface at a small enough scale look flat).
Suppose you now aim yourself to the left a bit, instead of straight downhill but more in the way of a traverse. The slope angle will be less than 45. As you continue moving your gaze to the left and up, counter-clockwise, the slope angle will be less and less until at some point you reach 0. Keep on going and you end up with a repetition in mirror image, just as when you stand on that microscopic spot and either look directly downhill to get 45 or directly uphill to also get 45 (it's not said to be a -45 slope when you look uphill or a +45 slope when you look downhill).
That zero slope direction, by the way, is the direction of the perpendicular to the normal of the coordinate system, where a flat XY plane with zero slope (another way of saying it is perpendicular to the normal) would cut the surface of the camel's hump.
Where people get confused by seemingly tricky surfaces like concave, convex or saddle points is they forget slope is computed to get the tangent plane, in effect looking at such small regions where seeing the surface as a tangent flat plane makes sense. However you calculate, the ultimate notion of "slope" as the tangent flat plane remains the same in all of them.