I love this Bhuz feed: "Can someone define Egyptian Shimmy, please?" The answers on this thread define "Egyptian" shimmies as "leg driven," the knee pushing back, relatively recent in Egyptian style dance, and one taught by the belly dance guru/teacher from Egypt, Raqia Hassan.
In class, we discussed the importance of staying as relaxed as possible through the quads while doing this shimmy. The weight is one the whole foot; there is a tendency to sink back into the heels, which, as in yoga, makes mobility in the hamstrings less fluid. The legs are alternating, the lower belly/pelvis is lifted (mulabandha helps if you do yoga), and the rib cage also lifts, which helps in fluidly isolating the upper and lower body. The shoulders and neck are relaxed. This shimmy is often done almost as a "resting shimmy,"and the dancer's upper body and arms are steady and free to express the music, balance a shamadan, play cymbals, etc. The video of Randa Kamal (above) is helpful because her body is very visible. This woman is so graceful and relaxed, the viewer almost misses how strong she is.
Here is a Youtube clip of two legends of Egyptian dance: Raqia Hassan and Mahmoud Reda: