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.
