It's the first one. This is a really excellent observation! It's a fascinating fact of physics.
Absolute potential energy is a silly idea. If you take a bunch of different objects, list their potential energies, and then add $100$ to each one, nothing will change about how the system behaves. We only talk about relative potential energy.
The kinetic energy an object gains in falling a certain height is equal to the potential energy it has lost. If we let an object fall from $h_1$ to $h_2$, we find that its change in kinetic energy is $\Delta KE = m g h_1 - m g h_2.$ If we add any arbitrary number $C$ to each of these potential energies, the difference is the same: $\Delta KE = (m g h_1 + C) - (m g h_2 + C) = m g h_1 - m g h_2.$
We often use the height off of the ground because it means that at ground level, $PE = mgh = mg \cdot 0 = 0$ for all objects, and since nothing can be lower than ground level in a simple system it makes sense to say that $0$ is the lowest possible potential energy anything can attain.
EDIT:
To add to this, let's look at a little extra mathematical formalism. It turns out that in classical mechanics, the nonexistence of "absolute potential energy" is a special case of something called gauge invariance.
For simplicity, let's talk about a one dimensional system - we have a ball that can only move back and forth along a single line. Let $x$ be the position of the ball.
Let $U(x)$ be the potential energy of the system as a function of the position of the ball. This could, for example, be a simple gravitational problem -- $x$ is the height of the ball above the ground, and $U(x) = m g x.$ But for generality, we won't specify what the form of $U$ is.
What we will say is that potential energy is the result of some force acting on an object. We know that the potential energy of an object at a certain position resulting from a given force is the work required to bring that object to that position. So if an object is acting under a force $F(x)$, the potential energy at position $x$ is $U(x) = W = - \int F(x)\, dx$ (the negative sign comes from the fact that we must work in the opposite direction of the force).
So from the fundamental theorem of calculus, if $U(x) = - \int F(x)\, dx$, then
$F(x) = - \frac{d U(x)}{d x}.$
Okay, this is interesting. In classical mechanics, we can completely describe the motion of a system if we know the forces acting on it (since we can then use Newton's law $F = ma$). But since we know $F(x) = - \frac{d U(x)}{d x}$, we can describe the motion of the system completely by knowing the potential energy.
Here's the payoff:
If the forces are the same for two different potential energy functions, then those potential energy functions result in the same physical behavior.
Mathematically:
If $U_1(x)$ and $U_2(x)$ are two potential energy functions such that $- \frac{d U_1(x)}{dx} = - \frac{d U_2(x)}{dx}$, then the potential energy functions result in the same physical behavior.
What does it mean if two functions have the same derivative? Well, it means that they differ by a constant.
Oh! That's where we wanted to get to, isn't it? If two potential energy functions differ by a constant, then they result in the same physical behavior. So it doesn't make sense to talk about "absolute potential energy", because no matter what we can add any constant we want and we'll obtain the same forces and thus the same physical behavior.
Hence it only makes sense to talk about changes in potential energy, not absolute potential energy.
(I said earlier that this is an example of a gauge invariance -- choosing a different constant to add to your potential energy function may be referred to as choosing a different "gauge" [which is a physical term]. The principle of gauge invariance states that the physical behavior of the system is the same regardless of which gauge you choose. In physics we often choose the gauge that makes our calculations the simplest -- which is why we choose the potential energy function $m g h$, where the potential energy is zero at ground level. This is an example of picking a useful gauge)
Best Answer
The potential energy has a gauge freedom, that is we can define the zero to be anywhere we want without affecting the physics. A side effect of this is that we cannot experimentally measure potential energy, we can only measure differences in potential energy.
So when you say the potential energy of an object raised to a height $h$ is $mgh$, what this really means is that raising an object by a distance $h$ changes the potential energy by $mgh$ i.e. the difference in the potential energy before and after raising was $mgh$.
The person holding the suitcase can define its potential energy to be zero, but this is just a choice of gauge. Regardless of how the person holding the suitcase defines the potential energy it still changes by $+mgh$ when it is raised by a distance $h$ and $-mgh$ if it is lowered by a distance $h$.