Lens makers strive to make lenses that deliver a faithful image. This has never been achieved, because all lenses suffer from aberrations (optical jargon for error). There are 7 major aberrations. To mitigate, lens makers construct a complex array consisting of several individual lens elements crammed into the lens barrel. Some are air-spaced from one-...
It means less distortion, less color aberration, less vignetting, a bit more sharpness.
This said the smartphone-ish look of your pictures could come from the sensor or the post processing and have nothing to do with the lens.
A lens that has a convex side and a flat side is called plano convex, so that would describe the sort of situation you're contemplating. Slides with wells exist to contain liquid samples; a convex version wouldn't be much use, so you won't find those manufactured. If you wanted to create such a thing, you could use some optical cement to glue a small plano ...
As you already suspected, it is most probably a partially smudged lens on that phone. It might also be a fingerprint just covering part of the lens. With tiny lenses, even the space between the ridges of a fingerprint can cause interesting effects.
You already had the right link there:
What causes lens flare along specific axes?