What you label as the "correct picture" - namely, the charge instantaneously losing potential as soon as it enters the resistor - is incorrect. Every resistive element in a circuit has a resistivity $\rho$, which is an intrinsic property of the material. The resistance of a resistive element with length $L$ and (assumed uniform) cross-sectional area $A$ is
$$R=\frac{\rho L}{A}$$
In other words, $\frac{\rho}{A}$ is the resistance per unit length of the resistor.
What this means is that our single resistor with resistance $R$ is equivalent to $N$ smaller resistors of length $\frac{L}{N}$ and resistance $\frac{\rho \frac{L}{N}}{A}=\frac{R}{N}$. If we were to analyze this circuit of $N$ smaller resistors, we would find that there would be a voltage drop $V=I\frac{R}{N}$ across each of them, for a given current $I$. This shows that the electrical potential energy of a charge depends on its position within the resistor. In fact, we can say even more than that: a charge that at position $x$ within the resistor (where $x=0$ is the entry point) has encountered a resistor of length $x$, and so the potential drop that that charge has experienced is, by Ohm's Law,
$$V(x)=I\frac{\rho x}{A}$$
In other words, this shows that the potential difference increases linearly with position within the resistor (equivalently, the potential energy a charge has decreases linearly with distance). This energy loss is usually described as coming from collisions with the electrons and atoms within the resistor. Given that the collision rate with electrons and atoms in a resistor should be constant with time and position (as the intrinsic properties of the resistor don't change with position inside it), the answer we're getting makes physical sense.
In circuit analysis, typically we don't care about what's going on inside a resistor; rather, we're more interested in how the resistor as a whole behaves relative to other circuit elements. As such, we tend to use the lumped-element model in circuit analysis, where the effects of the resistor happen all in one "lump" that we place in a network of zero-resistance "wires." Since we don't tend to connect other circuit components to a point in the middle of a resistor (and if we do, we usually use a special variable-resistor lumped element), this model works quite well, as it only obscures the details that we don't care about in that situation. However, if you do care about what happens in the middle of a resistor (as you do in your question), then you can't use the lumped-element model to predict the physics, because it wasn't built to do that.
