Two definitions of horofunction for Gromov hyperbolic spaces 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.
First Definition

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.
Second Definition
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 is 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:
  
  
*
  
*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 $\gamma\colon [0,1]\to X$.
  
*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 $\lambda\in (-\infty,f(x)]$.

My Question
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.
 A: I don't know whether this was known before, but Collin Bleak, Francesco Matucci, and I have settled this question in the course of our work on our recent paper [1].  The answer is that any horofunction satisfying Definition 1 satisfies Definition 2, but there exist hyperbolic groups with horofunctions satisfying Definition 2 and not Definition 1.
Definition 1 implies Definition 2
This is based on the following claim.
Claim. Each of the functions $f_p$ is $2\delta$-convex.
Proof: Let $\gamma\colon [0,1]\to X$ be a constant-speed geodesic of length $L$, and let $a=\gamma(0)$ and $b=\gamma(1)$.  Let $t\in [0,1]$, and choose geodesics $[p,a]$ and $[p,b]$ from $p$ to $a$ and $b$, respectively.  Since $X$ is $\delta$-hyperbolic, there exists a point $q$ on $[p,a]\cup [p,b]$ so that $d(q,\gamma_t)\leq \delta$.  If $q\in [p,a]$ then 
\begin{multline*}
d(\gamma_t,p) \leq d(q,p)+\delta = d(a,p) - d(a,q) + \delta \\
\leq d(a,p) - d(a,\gamma_t) + 2\delta = d(a,p)-tL+2\delta
\end{multline*}
so $f_p(\gamma_t)\leq f_p(a) - tL+2\delta$, and a similar statement holds if $q\in[p,b]$.  Thus
$$
f_p(\gamma_t) \leq \max\bigl(f_p(a)-tL,f_p(b)-(1-t)L\bigr)+2\delta
$$
for all $t\in[0,1]$, and it follows  that
$$
f_p(\gamma_t)\leq (1-t)f_p(a)+tf_p(b)+2\delta
$$
for all $t\in[0,1]$. $\square$
Taking a limit, we deduce that any horofunction satisfying Definition (1) is $2\delta$-convex.  The following claim finishes the proof.
Claim. Any horofunction satisfying Definition (1) is distance-like.
Proof: Let $\{p_n\}$ be an unbounded sequence of points in $X$ so that $f_{p_n}$ converges to a function $f\colon X\to \mathbb{R}$ uniformly on compact sets.  Let $x\in X$ and let $\lambda\in (-\infty,f(x))$, so $f_{p_n}(x) > \lambda$ for large enough $n$.  Since the sequence $\{p_n\}$ is unbounded, we also know that $f_{p_n}(p_n)<\lambda$ for large enough $n$.  For each such $n$, choose a geodesic $[p_n,x]$. By the Intermediate Value Theorem, there is a point $y_n$ on this geodesic so that $f_{p_n}(y_n) = \lambda$.  Note then that $d(x,y_n) = f_{p_n}(x) - \lambda$ for each $n$, so $d(x,y_n) \to f(x)-\lambda$ as $n\to\infty$.  Since $X$ is proper, the sequence $\{y_n\}$ must have a limit point $y$, which satisfies $f(y) = \lambda$ and $d(x,y) = f(x)-\lambda$.  Thus $d\bigl(x,f^{-1}(\lambda)\bigr) \leq f(x)-\lambda$, and the opposite inequality follows from the fact that $f$ is $1$-Lipschitz (since each $f_p$ is). $\square$
Definition 2 does not imply Definition 1
As Anton Petrunin suggests, it is sometimes possible to take something like a minimum of functions that satisfy Definition (1) to get a function that satisfies Definition (2) but not Definition (1).
For example, let $X$ be the Cayley graph of the group
$$
G = \langle a,b \mid ab=ba,b^3=1\rangle \cong \mathbb{Z}\times\mathbb{Z}_3
$$
where $x_0$ is the identity vertex, and note that $G$ acts on $X$ in a natural way. Let $T$ be the triangle of edges connecting $1$, $b$, and $b^2$.  Using Definition (1), each point $p\in T$ has two associated horofunctions, namely those associated to the sequences $\{a^n p\}$ and $\{a^{-n}p\}$, and it is not hard to check that these are all of the horofunctions on $X$ determined by Definition (1).  In particular, the horofunction boundary of $X$ by Definition (1) is homeomorphic to the disjoint union of two copies of $T$.
However, there are horofunctions for $X$ satisfying Definition (2) but not Definition (1).  For example, let $\{g_n\}$ be the sequence of functions
$$
g_n(x) = \min\bigl(d(x,a^n),d(x,a^nb),d(x,a^nb^2)\bigr) - n
$$
where $a^n,a^nb,a^nb^2$ denote the corresponding vertices in $X$.  Then $\{g_n\}$ converges uniformly on compact sets to a function $g\colon X\to\mathbb{R}$ which satisfies Definition (2) but not Definition (1).  In particular, $g(1)=g(b)=g(b^2)=0$, but this is not true for any horofunction satisfying Definition (1). More generally, for any values $u,v\in [-1,1]$ satisfying $|u-v|\leq 1$, there are exactly two Definition (2) horofunctions on $X$ that satisfy $g(1)=0$, $g(b)=u$, and $g(b^2)=v$.  The set of all such $u$ and $v$ is a closed hexagonal region in $\mathbb{R}^2$, and the set $\Phi$ of horofunctions satisfying Definition (2) is the union of two disjoint copies of this region.
Note that $G\times \mathbb{Z}_3$ will have similar behavior for any hyperbolic group $G$.  In particular, there exist non-elementary hyperbolic groups for which Definition (1) and Definition (2) are not equivalent.  Note also that the function $g$ defined above takes integer values on vertices, so it corresponds to a point in the space $\Phi_0$ of integral coycles defined by Coornaert and Papadopoulos.
