What is the reason for this difference between assumed and actual path variation?
Even your second image isn't correct. Imagine zooming in on a small portion of the Moon's orbit about the Sun, for example, one full moon to the next, with the Sun zoomed out of the picture. Now imagine drawing a line segment from one outer cusp (full moon) to the next. In both of your images, that line segment crosses outside of the curve. In other words, both of your curves are concave.
Compare that to the he Moon's orbit about the Sun. This is a convex curve. If you pick any two points on that curve and draw a line segment between them, the entirety of that segment will be on or inside the curve. The reason the Moon's orbit about the Sun is convex is because the gravitational force exerted by the Sun on the Moon is more than twice than exerted by the Earth on the Moon. The orbit would be concave if the Moon was closer to the Earth than 259000 km (about 40.6 Earth radii). Since the Moon orbits at about 385000 km (about 60.4 Earth radii), the Moon's orbit about the Sun is convex.
Whether the orbit of a moon about the Sun is non-simple (first image in the question), simple/concave (second image in the question), or simple/convex (Moon's orbit about the Sun), the deviations from an ellipse are tiny. With regard to the Earth-Moon system, the deviations are so very small that at the plotted resolution (288x288 pixels), the orbits of the Earth, the Earth-Moon barycenter, and the Moon about the Sun will be right on top of one another. The reason the variations are so small (less than one pixel at 288x288 pixels) is because of the huge ratio of the size of Earth/Moon orbit about the Sun compared to the size of the Moon's orbit about the Earth.
Those backward loops in your first image don't happen for any object orbiting the Earth. That would require an orbital velocity about the Earth greater than the Earth's orbital velocity about the Sun. The Earth's orbital velocity about the Sun is about 30 km/sec, considerably more than the orbital velocity of an object in low Earth orbit is about 7.8 km/sec.
Has this path been like this since the formation of the Moon?
No. The Moon formed at four to six Earth radii, far less than the 40.6 Earth radii figure cited above. The Moon's orbit initially looked like your second image.
Do Natural Satellites of other planets also follow the same orbit around the Sun?
The massive planets are much further from the Sun than is the Earth and are much more massive than is the Earth. The orbits of most of the moons of Jupiter about the Sun are concave rather than convex. Only the outermost moons of Jupiter have convex orbits about the Sun. A few of Jupiter's innermost moons (Metis, Adrastea, Amalthea, Thebe, Io, and Europa) exhibit the retrograde motion depicted in your first image.
With regard to moons whose orbit about the Sun is convex, the distances that correspond to the 259000 km value for the Earth are 129000 km for Mars, 24.1 million kilometers for Jupiter, 24.2 million kilometers for Saturn, 19.0 million kilometers for Uranus, and 32.3 million kilometers for Neptune. Both of Mars' moons orbit close-in. However, all four of the giant planets have moons whose semi-major axis orbit fall outside the corresponding limit.