I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. Below I deliver some definitions needed to explain my question.
Given a surface $\Sigma$ (i.e. a connected second-countable Hausdorff space locally homeomorphic to $\mathbb{R}^2$), a pre-end of $\Sigma$ is a nested sequence $\{U_{n}\}_{n\in \mathbb{N}}$ of connected open subsets of $\Sigma$ with compact boundary such that for all $n\in \mathbb{N}$, $U_{n+1}\subset U_{n}$ and for every compact $K\subset \Sigma$ there exists $l\in \mathbb{N}$ with the property that $U_{l}\cap K=\emptyset$.
Two pre-ends $\{U_{n}\}$ and $\{V_{n}\}_{n\in \mathbb{N}}$ are equivalent if for any $i\in \mathbb{N}$ there exists $j\in \mathbb{N}$ such that $V_{j}\subset U_{i}$ and vice versa. The set of equivalence classes of pre-ends is denoted by $E(\Sigma)$ and its elements are the ends of $\Sigma$. A topology is given to $E(\Sigma)$ by specifying a basis as follows: for any open subset $W\subset \Sigma$ whose boundary is compact, a basic open subset of $E(\Sigma)$ is given by $$W^*:=\{[U_{n}]\in E(\Sigma)\hspace{0.1cm}|\hspace{0.1cm}U_{l}\subset W \mbox{ for }l \mbox{ sufficiently large}\}.$$ The corresponfing topological space $E(\Sigma)$ is known as the end space of $\Sigma$.
A surface is said to be planar if all of its compact subsurfaces (possibly with non-empty manifold boundary) are of genus zero. An end $[U_{n}]$ is called planar if there exists $l\in \mathbb{N}$ such that $U_{l}$ is planar (equivalently, if there exists an open subset $W\subset \Sigma$ with compact boundary that is embeddable in $\mathbb{R}^2$ such that $[U_{n}]\in W^*$). Analogously, an orientable end can be defined.
Equivalently, the end space of $\Sigma$ can be described using proper rays (i.e. continuous maps $\gamma:[0,+\infty)\rightarrow \Sigma$ such that for every compact set $K\subset \Sigma$, the set $\gamma^{-1}(K)$ is compact as well) as follows: two proper rays $\gamma_{1}$ and $\gamma_{2}$ are equivalent ($\gamma_{1}\sim \gamma_{2} $) if for every compact $K\subset \Sigma$ there exist an $T>0$ such that $\gamma_{1}\big([T,+\infty)\big)$ and $\gamma_{2}\big([T,+\infty)\big)$ lie in the same path component of $\Sigma\setminus K$. In this setting, the end space of $\Sigma$ is the quotient $$e(\Sigma):=\frac{\{\gamma:[0,+\infty)\rightarrow \Sigma\hspace{0.1cm}|\hspace{0.1cm}\gamma \mbox{ is a proper ray}\}}{\sim},$$ equipped with the topology determined by the following convergence notion (see User's answer below for a description of a basis of such topology): If $end(\gamma)$ is the equivalence class of a proper ray $\gamma:[0,+\infty)\rightarrow \Sigma$, then a sequence of equivalence classes $\{end(\gamma_{n})\}$ converges to $end(\gamma)$ if for every compact set $K\subset \Sigma$ there exists a sequence of positive numbers $N_{n}$ such that, for $n$ sufficiently large, $\gamma_{n}\big([N_{n},+\infty)\big)$ and $\gamma\big([N_{n},+\infty)\big)$ lie in the same path component of $\Sigma\setminus K$.
The two previous definitions give rise to homeomorphic end spaces. Indeed, on one hand, given a pre-end $\{U_{n}\}$, we can create a proper ray by picking points $x_n\in U_{n}$ and connecting $x_{n}$ with $x_{n+1}$ using a path entirely contained in $U_n$. On the other hand, if $\gamma: [0,+\infty)\rightarrow \Sigma$ is a proper ray, we can construct a pre-end $\{U_{n}\}$ for which $\gamma\big([n,n+1]\big)\subset U_{n}$.
My question is whether or not the space of ends seen as a quotient of proper rays can distinguish the planar ends and the orientable ends.