In [this MO question](https://mathoverflow.net/questions/238310/competing-notions-of-%C3%A9taleness) I asked for some help with several definitions of *formally* etale maps. The definitions I'm asking about are 'isomorphism of tangent spaces', i.e the square below is a pullback
$$\require{AMScd} \begin{CD}
    TM @>{df}>> TN\\ @V{\pi}VV @VV{\pi^\prime}V\\
    M @>>{f}> N,
    \end{CD}$$
and the right lifting property against infinitesimal extensions, i.e for surjections with nilpotent kernel $\hat R\twoheadrightarrow R$ the existence of a unique diagonal filler
$$\array{
    \operatorname{Spec}R &\longrightarrow& \operatorname{Spec}B
    \\
    \downarrow &\nearrow& \downarrow
    \\
    \operatorname{Spec}\hat R &\longrightarrow& \operatorname{Spec}A.
  }$$

There's a [comment](https://mathoverflow.net/questions/238310/competing-notions-of-%C3%A9taleness#comment589778_238310) I would really like to unravel. It says:

> ... you should think of $\operatorname{Spec}\hat R$ as some kind of tubular neighborhood of $\operatorname{Spec}R$, so the property is saying that you can lift (small enough) tubular neighborhoods along local diffeomorphisms. If you think about it for a while you'll see that this is more or less equivalent to being a local diffeomorphism.

I thought about this for a while but didn't really get anywhere. I'm looking for:

 1. Geometric intuition (in the usual Euclidean world) to help me see why I should expect this equivalence intuitively.
 2. Formal statement. Which properties can actually be said to be equivalent?