You can't see clearly underwater for a couple of reasons. One is the thickness of your lens, but the main one is the index of refraction of your cornea.
For reference, here's the Wikipedia picture of a human eye.
According to Wikipedia, two-thirds of the refractive power of your eye is in your cornea, and the cornea's refractive index is about 1.376. The refractive index of water (according to Google) is 1.33. In water, your cornea bends light as much as a lens in air whose refractive index is
$$\frac{1.376-1.33}{1.33} + 1 = 1.034$$
That means you're losing about 90% of your cornea's refractive power, or 60% of your total refractive power, when you enter the water.
The question becomes whether your lens can compensate for that.
I didn't find a direct quote on how much you can change the focal distance of your lens, but we can estimate that your cornea is doing essentially nothing, and ask whether your lens ought to be able to do all the focusing itself.
For a spherical lens with index of refraction $n$ sitting in a medium with index of refraction $n_0$, the effective focal length is
$$f = \frac{nD}{4(n-n_0)}$$
The refractive index of your vitreous humor is about 1.33 (like water), and the refractive index of your lens, according to Wikipedia, varies between 1.386 and 1.406. Let's take 1.40 as an average. Then, plugging in the numbers, the effective focal distance of a spherical eye lens would be five times its diameter.
The Wikipedia picture of a human eye makes this look reasonable - a spherical lens might be able to do all the focusing a human eye needs, even without the cornea.
The problem is that your eye's lens isn't spherical. From the same Wikipedia article
In many aquatic vertebrates, the lens is considerably thicker, almost spherical, to increase the refraction of light. This difference compensates for the smaller angle of refraction between the eye's cornea and the watery medium, as they have similar refractive indices. [2] Even among terrestrial animals, however, the lens of primates such as humans is unusually flat.[3]
So, the reason you can't see well underwater is that your eye lens is too flat.
If you wear goggles, the light is refracted much more as it enters the cornea - the same amount as normal. If you want to wear some sort of corrective lenses directly on your eye like contact lenses, they should have a refractive index as low as possible.
Googling for "underwater contact lens", I found an article about contact lenses made with a layer of air, allowing divers to see sharply underwater.
How was this amount of zoom created by the water droplets on the concave lens of my glasses?
First, the typical magnification one gets from a drop of water is only a factor of ~4-5, not 10s to 100s. It is possible to construct micro-lenses from water droplets as discussed here https://www.nature.com/articles/ncomms14673, however that wasn't done to generate high magnification but rather to focus light from individual locations with a high spatial resolution.
...a single cell was zoomed in to the level where I could see the nucleus in the center, some squiggly organelles outside the nucleus and an irregular cell membrane around it.
Are you sure what you saw wasn't within or on the water droplet? There are lots of smallish bugs/animals in water and on your face/glasses that could be magnified by a water droplet. Since you state the source of the water was rain, it's more likely the small critter was already on your glasses (glasses can be rather nasty afterall).
There are lots of micro-animals like tardigrades, myxozoans, rotifers, nematodes, and loriciferans, many of which can look like a cell. I would guess this is a much more likely explanation than a single water droplet actually generating microscope-level magnification allowing you to see single cells, which are on the micron scale.
It is not possible to take a picture of this phenomenon. Probably the cells are in my eye.
This is unlikely. The amount of light that is reflected off of a water droplet vs transmitted is low (e.g., see https://en.wikipedia.org/wiki/Electromagnetic_absorption_by_water), which is why you can see through them.
It is possible you have some small organisms in your eye, but that the water droplet provides a magnified reflected image is unlikely. The most likely answer is the one I propose above, namely, that there are micro-animals on your glasses that are being magnified by the water.
The maximum angular resolution (e.g., see visual acuity) of the human eye is ~0.47 arc minutes or ~28 arc seconds. Typically ~1 arc minute is considered to be good vision. The smallest objects resolvable by the human eye are limited by the diffraction limit of the pupil and come in around the width of a typical human hair or ~55-75 micrometers (e.g., see https://en.wikipedia.org/wiki/Naked_eye).
So if we use the magnification of ~4-5 I mentioned above, something that is acutally ~0.055-0.075 mm in size would appear to be ~0.22-0.30 mm through an ideal water droplet. To put it another way, assuming the above and ideal conditions, the smallest resolvable objects through a water droplet would be ~0.011-0.015 mm.
With all that in mind, I note that the size range of single-celled organisms is rather large. There are bacteria that have cell diameters of less than a micron while some protozoans can be more than 100 microns.
Thus, in principle, it is possible that a large, single-celled organism was magnified and visible to your naked eye through the water droplet and your glass lense. However, I still maintain the more likely explanation is that there were some micro-animal-like critters on your glasses that were magnified by the water droplet.
Side Note
Depending on where you live in the world can also influence whether there is a significant probability of small micro organisms existing within the water drops from rain fall, i.e., they are inside the rain drops before they hit the ground. So I initially argue this is less likely above, it is not impossible for the rain water itself to contain micro organisms, e.g., see https://www.nature.com/articles/s41545-019-0030-5.
Best Answer
Yes. I am myopic and I see a slight double image along the edge of my glasses:
This means that the field of view inside the frame is bigger - zoomed out - than what it would be with the same frames but without the lenses.
Similarly, objects are slightly but perceptibly smaller than they are without the glasses, as is clear from the size of the mugs at bottom left of the image. Zooming in on them and putting the inside- and outside-the-glasses versions in direct contrast gives a clearer picture:
As has been pointed out, the effect is strongly dependent on your exact position with respect to the lens, and it is exaggerated in the picture above because the camera is artificially far from the lens. Visually, and with the glasses correctly worn, the effect is milder and it looks closer to the door handle below:
For the handle to look like this I need to be about 2m from the door; I have about 2.5 dioptres in that direction if my memory serves.
In practice, though, your field of clear vision is restricted to the frame of the glasses, which do not cover a lot of your peripheral vision. This means that in practice you have a much reduced line of sight compared with someone without glasses, or who uses contact lenses. Whatever amplification was gained through this mechanism - the optics of which have been described in detail in other answers - is completely lost to this, and it's something to be aware of with people wearing glasses - our field of view is rather restricted, regardless of whether one is myopic or hyperopic.