When using the multivariable chain rule to compute $dz/dt$ of something with the form $z = f(x(t), y(t))$, the geometry involves moving from a number line ($t$) to a plane ($x$, $y$) to a new number line ($z$). The equation $z = f(x(t), y(t))$ itself can be viewed as a progression of position vectors, or perhaps points to avoid later confusion, from the $t$ line to the $xy$ plane, to the $z$ line. These are shown as red points below. Similarly, $dz/dt$ can be viewed as a progression of tiny vectors stemming from these points, shown as relatively large purple arrows, but with the understanding that they're actually arbitrarily small. The dependencies are shown in the tree structure on the right.
How does this geometric picture change in the alternative case of computing $dz/dx$ of $z = f(x, y(x))$, the following tree structure?
One bizarre facet of this latter case is that its formula, $dz/dx = ∂z/∂x + (∂z/∂y)(dy/dx)$ reveals in clear terms that $dz/dx$ and $∂z/∂x$ aren't the same thing; the total change in $z$ with respect to $x$ isn't the same as the partial change in $z$ with respect to $x$.
My hypothesis is that the axis of the intermediate $xy$ plane in this case is replaced by something other than an orthogonal axis. The English clause "y is a function of x" or "y depends on x" sounds a lot like "y is influenced by x" or "y is partially determined by x," which aren't too unlike "a component of y points in the x direction." Consider a plane starting with standard basis vectors $î$ and $ĵ$ that is then transformed by $\left[\begin{array}{l}1&\sqrt{2}/2\\0&\sqrt{2}/2\end{array}\right]$. The transformed bases aren't orthogonal to each other anymore. Viewed in light of our original basis, we could say that a component of $y$ now points in the $x$ direction, which is very reminiscent of our multivariable calculus situation. This also shows promise for an analogy to the inequality of $dz/dx$ and $∂z/∂x$, as the basis vector that spans this plane's x-axis wouldn't be the full story of the $x$ components of the purple vectors. So a red point and purple vector would start on an $x$ number line and move to a $y$ number line, but from $z$'s perspective, its input wouldn't (in general) look like a Cartesian $xy$ plane, but a modified $xy$ plane, where the x-axis remains in place, but the y-axis is rotated, sheered, etc., perhaps even twisted in some non-linear way. It would stand to reason that iff $y$ is a linear function of $x$, the y-axis would be straight, and tools of linear algebra could be applied to shed new geometric insight into the problem.
Am I on the right track for thinking about the geometric picture of z(x, y(x)) and z'(x, y(x)), and if so, how can this be drawn out in a bit more detail?