TL;DR

Colder mass has more density and sinks while hotter mass has less density therefore it raises, but where exactly do the convection cycle begins?

Consider the following experiments.

First: heating soy oil in a glass kettle on a regular gas stove; 4 probes — at the bottom, midway, near the surface inside the oil, and in the air above the oil (as in the image below); turn the fire on and follow the changes in temperature.

Second: same stove-kettle setup, 3 probes (bottom, midway, and surface, as in the image below), ice on the surface and fire on simultaneously; here again my naive expectative was that the button, directly receiving the heat from the fire would be hotter than the surface, especially while having ice still floating on the surface.

I have obviously repeated the experiments with different containers, different types of probes, etc., and the results were qualitatively similar.

Could someone explain how this dynamic happens?

The videos of the experiments are at the links below:
Soy oil
Water & Ice

I understood that the convection cycle is at motion at the very moment the heat source is turned on, and that the motion upward and downward start also simultaneously, nevertheless, I still have a very basic doubt: is it the colder mass that displaces the hotter mass upwards or is it the hotter mass that displaces the colder mass downwards?

Both possibilities can result in a convection cycle, but their causality would be antagonistic.

Note that the internal movement or mixing depends on the location of the heat source.
Relocating the heat source to the surface harmonizes the internal forces induced by thermal energy gain and gravity: the hotter and less dense superficial mass tends to "float" above the colder mass and to remain in its original location.

Reading about stratification in nature may help you understand its oposite:
https://en.wikipedia.org/wiki/Stratification_(water)

You should read up on Rayleigh-Benard convection and the associated Rayleigh-Benard instability.

I know that this is not an exception that only happens in this specific setup because I have tested different types of containers, different types of liquids and several different types of probes/thermometers — and the qualitative outcome, i.e., heating from beneath makes the bottom colder; so, why is that?

This feature is barely at the limit of your experimental setup for oil. For the less viscous water and with the initial mechanical effects, it is not surprising that you miss it entirely there.

Here is a more detailed paper on the topic: https://courses.physics.ucsd.edu/2019/Winter/physics116_216/Berge and Dubois convection_W17.pdf

You should work through that paper, calculating the relevant quantities for your oil setup. In particular, the dimensionless parameters, and any length or time scales. I would focus on the oil due to the mechanical agitation of the ice

Could someone explain how this dynamic happens?

The videos of the experiments are at the links below:

Ice is floating in hot water because it has no other option, it is the less dense actor in there.

Simultaneously, it is absorbing heat from (and therefore reducing the temperature of) the water and the air that is surrounding it.

The recoded temperatures could be different if the three probes were located exactly along a vertical line.

In the following video, the white color represents water at higher temperature than the one represented by yellow.
Note how the most superficial layer is cooled by the air, and how the water melting from the ice immediately sinks.
In this case, the mass of ice/water rate seems to be smaller than the one in your experiment, which greatly reduces the melting time.

TL;DR: Colder mass has more density and sinks while hotter mass has less density therefore it raises, but where exactly do the convection cycle begins?

I’m having trouble understanding how exactly non-equilibrium heat-transfer initiates convection, whether or not phase change is involved.

Consider the following experiments.

First: heating soy oil in a glass kettle on a regular gas stove; 4 probes — at the bottom, midway, near the surface inside the oil, and in the air above the oil (as in the image below); turn the fire on and follow the changes in temperature.
View attachment 374222
I was expecting that the bottom would lead — i.e., the temperature at the bottom, specially during active heating, would be hotter than the temperature at the surface.
I plotted the dynamics on a chart:
View attachment 374223

Thats my take?

Roughly what I’m trying to understand is if the heat would simply be dissipated towards the cold butter and the butter would melt there and, obviously, spread itself through the pan from there, or if an active heat source could pull the butter towards it?

As in the red and yellow arrows, what would be expected: the red arrow — i.e., the hot from the flame reaches the butter and it melts there —, or the yellow arrow — i.e., when the heat reaches the butter, the butter is pulled/attracted by the heat?

Anyway, which physics specialty must be used to deal with this prediction was also one branch of my question; so forget the term thermodynamics for a moment and help me reason on the issue.

Assume a perfect flat leveled pan — the prediction from mainstream physics (whichever area of physics must be consulted) for the outcome should be:

Butter can slide any direction, whichever direction is downhill, regardless of which direction the heat source is. It could be towards or away from the hotter region or any other direction.

I totally understand your point and I partially agree: thermodynamics, as a field of physics, is not the (sole) determinant here — there will be other important correlate fields, e.g., solid conduction in steel, phase change in the butter, capillarity, wetting hysteresis, geometry, gravity, etc., —, nonetheless, I still beg to disagree, since the dynamics of the butter pat will be determined by the "thermo", but ok, I know I'm theoretically wrong (i.e., it's not about thermodynamics).

Anyway, which physics specialty must be used to deal with this prediction was also one branch of my question; so forget the term thermodynamics for a moment and help me reason on the issue.

Assume a perfect flat leveled pan — the prediction from mainstream physics (whichever area of physics must be consulted) for the outcome should be:
View attachment 374557
would the butter melts from its start position (red arrows) or would it be pulled towards the hotter region first and then melt there (blue arrows)?

Edit: I just ran your experiment here. The pat of butter only slid downhill. At no point did it slide uphill. Your result is not replicable.

This is one reason why we require all posts on PF to be consistent with the professional scientific literature.