So that's where the fun of Quantum Mechanics and uncertainty comes in. Electrons are not just point-like particles, they're sort of both particles and waves. If you nail down exactly where the electron "particle" is, then you end up with it's momentum being less defined, meaning that you're not going to be able to say anything at all about where it came from, where it's going, or how fast. Similarly if you can nail down it's momentum perfectly it won't have a definition of where it is anymore. This isn't the Observer effect, where you making the measurements changes the system, but in fact that the other related property actually ceases to have a valid definition anymore (I'm not well versed enough to say it stops existing with any confidence but that's my understanding). This leads to a situation where you end up measuring a probability density of the property of the electron, so you get a density of it being "around this point +/- some amount" and with a "momentum +/- this amount, and sort of in this direction". This means that you now have a probability density of the electron where you'll see it's effects of the negative charge and kinetic energy and those are the things that describe that "cloud of negative charge". The main effect of this that affects things is that you end up with this cloud of all the electrons around an atom and you don't ever observe the individual electrons, just the shells/clouds around the atom and how they change as energy is inserted into the system.
My layman's understanding is that essentially as you make a measurement/system/whatever where it's designed to see the position more than the momentum you'll see more point-like particle behavior, but never only just that (you can't ever nail it down to an infinite precision, if you did it'd have a momentum that could send it off at the speed of light as if it had no mass) and in the other case if you could measure infinite precision on the momentum you'd end up measuring the electron as if it was "everywhere" in the universe all at once, which is just as nonsensical of a result as the first one.
So for the simplest explanation for you and the parent is that as you said electrons are both particles and waves, there are experiments that we can demonstrate for that. But if you're thinking of the electron cloud then that exists partially because you aren't measuring the electron. Basically this particle(s)/wave(s) orbiting the atom can exist in a lot of different configurations at once determined by it's energy (which is influenced by its/they position and how fast it is/they are moving). For educational purposes we often treat them more exclusively as a particle (early electrodynamics) or a wave (early quantum) depending on what field we are talking about.
Of course there is greater definition I can give to what it is if we want to start talking about what protons and neutrons are made of and most of that is what occurs at the LHC. (But I'm heavily out of my depth in high energy particle physics).
The measurement uncertainty applies to some other properties as well basically your uncertainty in the electron's position (sigma_x) and uncertainty in the electron's momentum (sigma_p) are bound by sigma_p * sigma_x >= hbar/2
hbar is the reduced planck constant and sigma are statistical variance, but that condition must always hold so as sigma_x goes to 0 (you are more certain about the potential) sigma_p must get larger to compensate so you are less certain about the momentum. This measurement collapses the waveform and until enough time passes for the system to "normalize" the momentum will continue to be uncertain, after that you can measure the momentum but the position could be different too.
That's a small detail, but no, it's not that the particle "stops existing". It's that we can't describe the momentum and position of something with indefinite detail with mathematics that is still coherent. The data type we have for describing things can not go there (on the QM language, of course).
And the surprising thing is that QM predicts our experiments with as much detail as we can experiment with. So, we have this language that can't describe something with known position and speed, and as far as anybody is concerned, it's perfectly correct.
Such a quantized vibration has all the properties of a particle: it has a certain mass, it has a certain momentum, it has a certain angular momentum, and when you look at it in another frame of reference, all these quantities transform just as would a particle’s. And, when you “perform a position measurement” it shows up as a single dot. If you don’t do a position measurement it spends most of its time quantum superposition of states of definite position; hence the “cloud” aspect.
Similarly, a non vanishing Electric Dipole Moment would be some preferred direction to its electric interactions in relation to its spin. Its electric field would be slightly oblong even when position is localized to a dot.
So to clarify, there are multiple senses of “electron cloud” at play — the superpositions of positions is what is usually meant by the term, as in electron cloud around nucleus of atom. The article’s use of the term is a tad sloppy since it invites confusion with this — it means something that generates an oblong electric field and reacts to an external electric field through a term in the energy of the form p.E (with p the EDM and E the external electric field).
If we forcefully interact with electron we can reshape this "cloud" to be as small as we need to at the cost of imparting on it random monentum through uncertainty principle.
I think what they measure is that when you interact with electron then its electric field seems to be very spherical to the point that we can rule out a lot of ideas about electron having any internal substructure that might cause the field to be a little bumpy.
I think it's worth mentioning it's not exactly a probability understood as a theoretical chance of something being somewhere. This wave function is what actually is the physical reality. But we can't measure this reality without interacting with it and when we do interact multiple times and interpret those interactions as if they were caused by pointlike particle being in particular spot then we see that results of those interactions have distribution matching this "probability" "cloud" (square of a wave function).
As more and more electrons get stuck to nucleus it gets crowded around it so further electrons form more and more bizarre shapes of larger and larger sizes as only two electrons can stably fit in a single specific shape of a specific size (Pauli exclusion principle). They do it by taking opposite spins.
You can look at simulated orbitals here: https://falstad.com/qmatom/
s-orbitals are spherically symmetrical. You might convince yourself that p-orbitals are also fine, because 3 of them in 3 dimensions do have reasonable rotational symmetry.
You'll notice that when you move to d-orbitals things start to get really weird. 4 of them are fairly equivalent but it turns out that you can cram 10 electrons of this energy around the nucleus in this fashion. So there are 5 d orbitals. And we can't really fit the fifth in similar manner as other 4.
https://profound-advice.com/what-is-the-difference-between-t... Check the video.
So while we technically have 5 d orbitals, one of them is weird. Its weirdness is an artifact of math. To get the orbitals we are solving Schrodinger equation in spherical coordinates, where one axis is chosen as special. This gives us this weird solution. https://chemistry.stackexchange.com/questions/117243/why-is-...
Physical reality can't depend on our choice of coordinate system so this tells us that orbitals as we draw them don't really directly express the physical reality. In actual physical reality all five of d orbitals have the same shape (somewhere in between the nice d and weird d), but since we can't tell the difference between electron on one d orbital from the electron on another then as long as all 5 orbitals sum up to physical reality our math still works, even if we choose to decompose the solution into 5 non-rotationally-symmetrical parts to make math easier on ourselves.