Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x_0$ be a basepoint for $X$.  There seem to be two different definitions of "horofunction" for $X$, and I'd like to understand the relationship between them.

<h3>First Definition</h3>

> **Definition 1.** For each $p\in X$ let $f_p\colon X\to\mathbb{R}$ be the function
$$
f_p(x) = d(x,p)-d(x_0,p).
$$
A function $f\colon X\to \mathbb{R}$ is called a ***horofunction*** if there exists an unbounded sequence $\{p_n\}$ in $X$ such that $f_{p_n}$ converges uniformly to $f$ on compact sets.

This definition is due to Gromov, and the set of all horofunctions on $X$ is known as the ***horofunction boundary***.  Note that this definition works for any metric space.

<h3>Second Definition</h3>

The following definition seems to come out of the work of Coornaert and Papadopoulos on the symbolic dynamics of the visual boundary of a hyperbolic group, though it similar to the "local" description of horofunctions using cocycles given by Gromov in his essay on hyperbolic groups.

> **Definition 2.** A function $f\colon X\to \mathbb{R}$ with $f(x_0)=0$ is called a ***horofunction*** if it satisfies the following conditions:

> 1. There exists an $\epsilon>0$ so that $f$ is $\epsilon$-convex, in the sense that
$$
f(\gamma_t)\leq (1-t)f(\gamma_0) + t f(\gamma_1) + \epsilon
$$
for every constant-speed geodesic $\gamma\colon [0,1]\to X$.

> 2. The function $f$ is distance-like, in the sense that
$$
f(x) = \lambda + d\bigl(x, f^{-1}(\lambda)\bigr)
$$
for every $x\in X$ and every $\lambda\leq f(x)$.

<h3>My Question</h3>

What, exactly, is the relationship between these two definitions?  Are they equivalent?  Is the second a generalization of the first?  I'd particularly appreciate a reference to a paper that discusses both definitions.