I've spent some time over the last few days working on this.
There is good reason to define a diameter as the locus of the midpoints of parallel chords.
This is because the definition as "chord which passes through the center" is inadequate as a definition for the following reasons:
a) It does not encompass the case of the parabola, where it is straightforward to demonstrate that the locus of the midpoints of parallel chords is a straight line parallel to the axis of the parabola. Thus, and because there is no center of a parabola (unless you go down the rocky road of defining the center as the point at infinity, and there is no real need to do that here) it is meaningless (or at best cumbersome and complicated) to define a diameter in relation to such a point.
b) In the case of the hyperbola, it does not take account of the diameters which are the loci of parallel chords either ends of which are on opposite branches of the hyperbola.
Hence the "chord through center" definition misses these important diameters which do not intersect the hyperbola at all.
So the question is resolved. The reason for defining a diameter as the locus of the midpoints of a system of parallel chords is because it is universally applicable.
The fact that a diameter passes through the center of the (central) conic section is a direct consequence of the above definition.
It would be so nice, though, if more texts (and websites) took the trouble to explain the above.
It would also be nice if the websites would construct adequate and accurate diagrams illustrating the concept without forcing the readers to scratch their heads and say: "But those are not the midpoints ..."
You asked
Because of this, is there a model of hyperbolic geometry that resolves all of these "problems" (i.e. it uses Euclidean metric, a negatively-curved surface, simple geodesics/lines, and a familiar geometric framework), or is it impossible?
and the simple answer is it is not possible. The reason is that
what you want can only be done for a geometry that has greater
curvature than the space it is embedded in. Thus, in Euclidean
space of zero curvature we can have nice models of positively
curved surfaces. If we lived in a negatively curved space, then
we could have nice models of surfaces of greater curvature such
as the Euclidean plane. However, we still could not have nice
models of surfaces with more negative curvatures.
One reason, among others, for this situation is that
the perimeter of a circle expands linearly as the radius
increases in Euclidean spaces. In hyperbolic spaces,
the perimeter expands essentially exponentially. Thus,
there is no room in Euclidean space to contain all of
the perimeter in a nice way without compromises.
Best Answer
Rectangular hyperbola is given by $x^2-y^2=1$. Consider a diameter given by $y=kx$ (for relevant $k$). In order to describe its conjugate we need to find the midpoints of the chords parallel to $y=kx$. Such a chord is given by $y=kx+n$, so let $(x_1,y_1)$ and $(x_2,y_2)$ be the endpoints of this chord. Note that we are interested only in the midpoint of the chord, i.e. in $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$, hence we don't really need to calculate the coordinates but only $x_1+x_2$ and $y_1+y_2$.
Endpoints of the chord are in the intersection of $x^2-y^2=1$ and $y=kx+n$. If we solve it for $x$, we get $x^2-(kx+n)^2=1$, i.e. $(1-k^2)x^2-2knx-n^2-1=0$. The solutions of this equation are $x_1$ and $x_2$, so by Vieta's formulae we have $x_1+x_2=\frac{2kn}{1-k^2}$.
If we solve the system for $y$, we have $(\frac{y-n}{k})^2-y^2=1$, i.e. $(y-n)^2-k^2y^2=k^2$, and we get $(1-k^2)y^2-2ny+n^2-k^2=0$. As before, $y_1+y_2=\frac{2n}{1-k^2}$.
So the midpoints of the chords are $\{(\frac{kn}{1-k^2},\frac{n}{1-k^2})\mid n\in\mathbb R\}$. This is obviously a line given by $y=\frac{1}{k}x$, which is obviously symmetric to $y=kx$ with respect to an asymptote.