Recall two connections on a manifold are said to be projectively equivalent, if they have the same geodesics. You want to know what local diffeos $\mathbb{R}^n\mapsto\mathbb{R}^n$ preserve geodesics; that is, what flat metrics on $\mathbb{R}^n$ are projectively equivalent to the Euclidean one. I think your conjecture, that the only such metrics are those obtained by affine-linear transformations of the Euclidean metric, is correct.

Let's discuss this first locally, then globally.

Local question: What flat connections on a neighbourhood of $\mathbb{R}^n$ are projectively equivalent to the Euclidean one?

Answer: If $n=1$, all. If $n\geq 2$, exactly those obtained by what depending on your terminology you might call a a perspective transformation or a projective linear transformation or something else.

Comment: Sketch proof below. The case distinction (which explains your colleague's observation) comes from a factor of $1/(n-1)$ in the formula for the appropriate Schouten curvature tensor. This is analogous to the case distinction $n\leq 2$ vs $n\geq 3$ in conformal geometry, see http://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)

Now we can deal with

Global question: What flat connections on $\mathbb{R}^n$ are projectively equivalent to the Euclidean one?

Answer: Just the standard one. The others "blow up"/"go off to infinity"/involve-division-somewhere-by-zero if you try to extend them to all $\mathbb{R}^n$.

Sketch proof of local version: Use the notation and some formulae from, eg., http://www.maths.adelaide.edu.au/michael.eastwood/projective.pdf

Let $\nabla$ be the standard connection on $\mathbb{R}^n$. Consider a projectively equivalent connection defined by a 1-form $\Upsilon_i$ (so the new connection acts on a 1-form $\omega_i$ by, $\nabla_i\omega_j - \Upsilon_i\omega_j -\Upsilon_j\omega_i$.) The projective Schouten curvature tensor of $\nabla$ is zero (since it's flat). If the new connection's also flat, then, applying appropriate transformation laws, we have that $\nabla_i\Upsilon_j = \Upsilon_i\Upsilon_j$, and that $\Upsilon_i$ is closed, so locally exact.

Write $\Upsilon = df$. Then in standard Euclidean co-ordinates we have the system of PDE $\partial_i\partial_j f = \partial_i f \ \partial_j f$ for the function $f$, which we can solve to get $f(x^1, ... x^n) =-\log (a_1 x^1+ \cdots + a_nx^n + c)$ for some fixed constants $a_i$ and $c$. Hence $\Upsilon_i = \frac{-a_i}{a_1 x^1+ \cdots + a_nx^n + c}$. It should probably turn out that the family of connections this gives are all indeed flat, and correspond to the projective-linear-transformations of the Euclidean metric on $\mathbb{R}^n$.

Recall two connections on a manifold are said to be projectively equivalent, if they have the same geodesics. You want to know what local diffeos $\mathbb{R}^n\mapsto\mathbb{R}^n$ preserve geodesics; that is, what flat metrics on $\mathbb{R}^n$ are projectively equivalent to the Euclidean one. I think your conjecture, that the only such metrics are those obtained by affine-linear transformations of the Euclidean metric, is correct.

Let's discuss this first locally, then globally.

Local question: What flat connections on a neighbourhood of $\mathbb{R}^n$ are projectively equivalent to the Euclidean one?

Answer: If $n=1$, all. If $n\geq 2$, exactly those obtained by what depending on your terminology you might call a a perspective transformation or a projective linear transformation or something else.

Comment: Sketch proof below. The case distinction (which explains your colleague's observation) comes from a factor of $1/(n-1)$ in the formula for the appropriate Schouten curvature tensor. This is analogous to the case distinction $n\leq 2$ vs $n\geq 3$ in conformal geometry, see Wikipedia - Liouville's theorem (conformal mappings)

Now we can deal with

Global question: What flat connections on $\mathbb{R}^n$ are projectively equivalent to the Euclidean one?

Answer: Just the standard one. The others "blow up"/"go off to infinity"/involve-division-somewhere-by-zero if you try to extend them to all $\mathbb{R}^n$.

Sketch proof of local version: Use the notation and some formulae from, eg., http://www.maths.adelaide.edu.au/michael.eastwood/projective.pdf

Let $\nabla$ be the standard connection on $\mathbb{R}^n$. Consider a projectively equivalent connection defined by a 1-form $\Upsilon_i$ (so the new connection acts on a 1-form $\omega_i$ by, $\nabla_i\omega_j - \Upsilon_i\omega_j -\Upsilon_j\omega_i$.) The projective Schouten curvature tensor of $\nabla$ is zero (since it's flat). If the new connection's also flat, then, applying appropriate transformation laws, we have that $\nabla_i\Upsilon_j = \Upsilon_i\Upsilon_j$, and that $\Upsilon_i$ is closed, so locally exact.

Write $\Upsilon = df$. Then in standard Euclidean co-ordinates we have the system of PDE $\partial_i\partial_j f = \partial_i f \ \partial_j f$ for the function $f$, which we can solve to get $f(x^1, ... x^n) =-\log (a_1 x^1+ \cdots + a_nx^n + c)$ for some fixed constants $a_i$ and $c$. Hence $\Upsilon_i = \frac{-a_i}{a_1 x^1+ \cdots + a_nx^n + c}$. It should probably turn out that the family of connections this gives are all indeed flat, and correspond to the projective-linear-transformations of the Euclidean metric on $\mathbb{R}^n$.