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What is the fundamental difference between thinking of models and the plane of a tangent to a curve?

The setting for the problem is the standard notion of a curve with curvature $k = \displaystyle \frac{d^2x^{\mu}}{ds^2}$. There is the notion of a the representation or parametrisation, $\phi(s) = (x(s),y(s))$ and then there is the notion of the tangent and normal. But the main difficulty of the problem lies in how they are related.
If one is told that there is a plane $P$ tangent to the curve at $s_1$, then presumably $s_1$ is the point where $\phi(s) \in P$. But then one is forced to suppose that there is a general representation where a tangent space is a necessary condition. And then if one notices that $P$ is not in the image of $\phi$, then one has to make a choice as to which tangent space. But if there were a curve which wasn’t locally one dimensional, one would have to make a choice as to which curve to use.
But what is the fundamental difference between these two? I understand the intuition that for a curve in three-dimensional space, the vector field of the derivative must be orthogonal to the curve but beyond that, the two notions seem to be completely equivalent to the first. One might suppose that the reasoning is that $n(s)$ is somehow attached to $P$, but this seems unsatisfactory since it requires one to choose a specific representation of a curve.
I have seen two different explanations: one is that the tangent at $s_1$ is $\phi(s_1) + t(s_1)$, but I would say this implicitly assumes that the tangent field is a vector field, since it is presumably normalised, and therefore it is attached to the curve at $s_1$. And in a sense that makes sense since the representation only has one vector at a time. But this is confusing.
The second explanation is that in a neighbourhood of

And this is the error I get when I run x3gvc again:

arackx3gvc: usage: x3gvc [options] [file]
arguments must be one or more X3GuvcExports

How can I get x3gvc to work?

A:

I found the answer elsewhere. It was something with the way I moved files.
I had a Windows 7 VM with ESXi 4.1. I had ESXi 4.1 Developer Edition installed on it. I moved the files from the Windows 7 VM to the ESXi host but I did not copy the folder that contained the Virtual Machine from the Windows 7 VM to the ESXi host.
I added the directory that I copied from the Windows 7 VM to the ESXi host to the PATH.
And that’s it. Now the x3gvc.exe works.

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