Difference between electric field $\mathbf E$ and electric displacement field $\mathbf D$

$\mathbf E$ is the fundamental field in Maxwell equations, so it depends on all charges. But materials have lots of internal charges you usually don't care about. You can get rid of them by introducing polarization $\mathbf P$ (which is the material's response to the applied $\mathbf E$ field). Then you can subtract the effect of internal charges and you'll obtain equations just for free charges. These equations will look just like the original Maxwell equations but with $\mathbf E$ replaced by $\mathbf D$ and charges by just free charges. Similar arguments hold for currents and magnetic fields.

With this in mind, you see that you need to take $\mathbf D$ in your example because $\mathbf E$ is sensitive also to the polarized charges inside the medium (about which you don't know anything). So the $\mathbf E$ field inside will be $\varepsilon$ times that for the conductor in vacuum.

Like @Marek has said above, the electric field $E$ is the fundamental field, and is in some sense the more physical. However, Maxwell's equations have a neater geometric meaning if you throw in the "auxilary" fields $D$ (and $H$ for $B$). I usually tell my students the following version of electromagnetism:

There are 4 fields in electromagnetism. We call them $E$, $D$, $B$ and $H$. All of these fields are independent and equally important. Furthermore, they actually embody geometric concepts which are manifest in the integral equations: $$\oint_S D \cdot dS = Q(S)$$ $$\oint_S B \cdot dS = 0$$ $$\oint_{\partial S} E \cdot dl + \partial_t \int_S B \cdot dS = 0$$ $$\oint_{\partial S} H \cdot dl - \partial_t \int_S D \cdot dS = \int_S j \cdot dS$$

Note that:

1. $E$ and $B$ form an independent pair, as do $D$ and $H$.
2. $E$ and $B$ do not depend on the sources $Q$ and $j$, but $D$ and $H$ do.
3. $D$ and $B$ are integrated through surfaces, and represent flux through those surfaces. (The correct mathematical gadget to describe these are actually 2-forms.$
4. $E$ and $H$ are integrated along lines, and end up representing the potential difference across the ends (or circulation in a loop).
5. The latter pair connect the change of flux through surfaces with certain circulations.

These equation form Maxwell's equations. They do not uniquely determine a physical situation. In particular, they need to be augumented with constitutive relations which describe (macroscopic) material properties. For example, we might have linear, isotropic, homogeneous (LIH) media, in which case we would have $D = \epsilon E$ and $B = \mu H$. But in general, we might have $\epsilon$ and $\mu$ being tensors, varying as functions of time and space, or even depend on the fields $E$, $B$, etc! These constitutive relations could be arbitrarily complicated, and indeed much of the new field of meta-material engineering is all about creating micro-structures which would yield interesting and useful constitutive relations at the macroscopic scale. More commonly, a scenario where the linearity breaks down is in ferromagnets/ferroelectrics.

There is usually another constitutive relation, linking current and electric field. In LIH media this is called Ohm's Law: $J = \sigma E$.

There is one more equation, which is simply always true, which is the conservation of charge; in the notation above, $\partial_t Q(S) - \int_S j \cdot dS = 0$.

Edit: some additional observations:

In a relativistically covariant form, we can merge $E$ and $B$ together to get the 2-form $F$, and $D$ and $H$ to get its Hodge dual $\star F$. The latter in general depends on the metric we choose. For linear materials it's possible to hide the effects of the material polarisation/magnetisation as a background metric. Incidentally, in this form, the energy is given by $F \wedge \star F$, so it is clear that energy/momentum should be "opposing" pairs, i.e. the Poyntin vector is $N = E \times H$.

In numerical simulations, it's doubly important that we obey Maxwell's equations --- failure to do so leads to highly unphysical things like superluminal propagation of waves or failure to conserve energy or momentum. It has been found that the key is to be exact with respect to the integral forms of the equations, and put all of the discretisation error into not meeting the material constitutive properties.

The electrical field $\mathbf E$ is the fundamental one. In principle, you don't need the electrical displacement field $\mathbf D$, everything can be expressed in terms of the field $\mathbf E$ alone.

This works well for the vacuum. However, to describe electromagnetic fields in matter, it is convenient to introduce another field $\mathbf D$. Maxwells original equations are still valid, but in matter, you have to deal with additional charges and currents that are induced by the electric field and that also induce additional electric fields. (More precisely, one usually makes the approximation that the electric field induces tiny dipoles, which are described by the electric polarization $\mathbf P$.) A little bit of calculation shows that you can conveniently hide these additional charges by introducing the electrical displacement field $\mathbf D$, which then fulfills the equation

$$ \nabla· \mathbf D = \rho_\text{free} .$$

The point is that this equation involves only the "external" ("free") charge density $\rho_\text{free}$. Charges that accumulate inside the block of matter have already been taken into account by the introduction of the $\mathbf D$ field.