While I have heard that a lower bottom bracket tends to give more stability, I'm confused as from which point of reference yields such an effect, from the wheel axle (BB drop) or from the tire contact patch (BB height)?
The conventional wisdom is you get more stability the closer you can get the center of mass to the contact patch. There are many other factors that could play a role in this than just the ones related to BB height (such as cargo placement), but the reason BB height matters so much by this reasoning is that the rider's body will typically be in a fixed position relative to the BB center. In other words, the rider positioning/ergonomics are conceptualized as not being affected by the BB height, but they are offset up and down in space by it. Thus, the ground and the BB center are the reference points.
BB height and BB drop are numbers with different meanings and uses that are both used to work with this concept.
BB height is the measure of the ground to the BB center given a specific set of tires. Thus, a BB height number by itself tells you everything about the BB's relationship with the ground, but contains no information about the drop number or the inflated diameter of the tire. (If you had one of those numbers additionally, you know the other.)
BB drop is the measure of the BB center's vertical distance away from a line drawn between the axle centers with a given headset. Thus if you know it, you know what the BB height will be with a tire of any inflated diameter. (You could also take it a step further and add the effect of rim width to the inflated diameter.) With a drop number you can answer questions like what is the smallest tire you can run while maintaining a target minimum BB height or what is the largest you can run without surpassing a target maximum. Framebuilders concerned with getting to the lowest BB possible while avoiding pedal strike will decide on a drop based on a chosen crank length, whereas mass-produced frames tend to have conservatively high BBs (low drop) to minimize the risk of pedal strike with a variety of crank lengths.
BB drop is a more straightforward number for comparing the design intentions of different frames. Using BB height number for the same purpose requires stipulating what tire was used to get there, because BB height contains no information by itself about the tire diameter.
There is a deeper conversation regarding some of the assumptions made in the premise of "lower center of mass = more stability." One is when and whether the rider position actually should be conceptualized as immutable in relation to the BB center. On inclines, in order to maintain traction a rider may need to shift their mass relative to the ground, leaving behind the notion that their mass is fixed relative to the BB center were they on a different bike with a lower BB. In turn, between two different frame designs that chose different BB heights for ground clearance needs, the higher one could have longer chainstays to mitigate this effect in the climbing direction, further muddying the viability of simple comparisons that only look at BB height/drop as a factor. These considerations are likely to weigh more on frame designs for technical as opposed to high-mileage, load-carrying, or recreational/utilitarian riding.
As others have stated, both numbers express how low the rider's/bike's center of gravity is, so they both relate to stability.
Bottom bracket height inherently will vary with tire size. To provide BB height, the geometry chart would have to specify some sort of reference tire. The problem is that nominal tire sizes still vary in width and diameter when mounted, i.e. a 25mm Conti GP 5000 and a 25mm Michelin Pro Cup probably have different diameters. Moreover, the tire's size and diameter will vary depending on the rim they're mounted to, with wider rims (I think) increasing the diameter.
Furthermore, all-road bikes or road-oriented gravel bikes can be designed to accept up to, say, 40mm tires, but they can still perform nicely with 28-32mm road tires mounted. This would cause a pretty big change in BB height. With older road frames that would typically come with 25mm tires, with clearance for 28mm, the BB height might not vary as much.
Bottom bracket drop is what frame makers specify since bottom bracket height depends on the tire size.
However, what you should be using is the bottom bracket height. That's the parameter you're interested in, it specifies cornering clearance and stability. However cornering clearance also depends on crank length.
Usually for good cornering clearance on 700c (622mm bead seat diameter) wheels, BB drop is about:
215 - crank length + tire width
So for example on my Surly Long Haul Trucker where I use 170mm cranks and 28mm tires:
215 - 170 + 28 = 73
However, the Surly LHT has 78mm BB drop, so one can see it doesn't have practically any cornering clearance even for the 170mm cranks. The reason is that I'm misusing it; as a touring bike it's intended for larger tires. Actually I think the tire size ideal is 35mm for a touring bike and it has been designed to be usable for 175mm cranks too. Let's recalculate:
215 - 175 + 35 = 75
So it has 3mm more bottom bracket drop than what good cornering clearance would need. That's because it's a touring bike. A general purpose road bike would remove 3mm from the BB drop of this frame (plus assume maybe 23-32mm tires instead of 35mm tires, meaning BB drop would need to be much smaller actually).