Different notions of isometry for Riemannian $2$-manifolds There are two notions of isometry between Riemannian $2$-manifolds: 


*

*a distance-preserving map $f$ with $d(x,y) = d(f(x),f(y))$ and

*a "metric-preserving" map $f$ with $I(x) = I(f(x))$ ($I(x)$ being the first fundamental form)


The second isometry surely implies the first, but what about the other direction? Can there be metrics and isometries (in the first sense) that don't imply isometries in the second sense?

When the metric is preserved in a map, the lengths and angles are both preserved. In a differential triangle constant lengths imply constant angles as well, in mapping ( Cosine trig rule).
For distance only preserving map. As an example compare one patch of Chevbychev net (described by Sine-Gordon pde) with the other. Each side length of differential rhombus element is same, but not the included angle which increases towards the cuspidal equator as shown.

Arc distances not on the net viz., circumferential diagonals can increase and longitudinal diagonals can decrease in length.
This is not full isometry.
Please note what Gauss Egregium theorem says in other words:
In full isometric mapping, i.e., when angles and lengths are both faithfully mapped, Gauss curvature K is conserved.
But in a converse situation when K is conserved in  mapping we can have
either lengths conserved and angles varying.. as well known in the fishing net / Chebychev net  that maps differential cells within the surface
or angles conserved ( conformal maps) and lengths varying.
So I would like to coin the term " iso-lineal " mapping where lengths are preserved but not the angles included between them for Fishing net or Chebychev net.
EDIT 1:
After the post I realized I missed the possibility ... that Loxodromes on spheres or pseudospheres can have patches in them mappable with constant angles (conformal mapping), and lengths of differential quadrilateral sides  (parallelogram sides) varying in length.
A particular case  is computed and shown with Curvilinear Parallelograms. Curvilinear rectangles and squares are other special cases.

Arc parametrization
$$ \{r,\theta, z \}=  \{u(s),v(s), w(s)\} $$
is possible in closed form with elliptic integrals. Angles are preserved but sides are dilated/shrunk. I believe particular case is  relevant to your question.
