The Cancer symbol (also called Cancer glyph) is a shorthand representation of the Cancer zodiac animal (see below). However, as with the animal itself, much is based both in interpretation and the preference of the writer, and can range from simple to extravagant.
Above are some examples of commonly used styles for the Cancer symbol. Each of these is equally valid - there is no 'correct' version as long as the general shape is maintained.
The word 'Zodiac' means circle of animals (though note a couple of signs aren't animals in the normal sense). These 'animals' were based originally on the pattern of the stars in their corresponding star constellation. Basically what the earliest astronomers and astrologers saw when they squinted at the night sky.
Due to this although Cancer will always be represented by a crab, there isn't any one 'correct' shape or design for that crab. It's a cosmic game of join the dots, and some constellations even have alternative ways to form the animal from the dots (stars).
Any picture of the shape, head or profile of a crab can therefore be comfortably used here. Designs range from the space-age to the ancient, and are limited only by the imagination of the artist, as long as it gets the point across.
Below is a representation of Cancer (and the Cancer symbol, inset) from a medieval wood carving from the 16th century.
As can be seen, it's a picture of lobster! The roots of modern astrology go back to the Babylonian/Assyrian empire around 500bc, and as with anything that old translation and re-translation causes some issues. Per the literal translation, Cancer refers to a creature which carries it's home on it's back, has lots of arms, and often travels sideways.
Clearly they weren't very picky about their seafood back then! At some point over the past 400 years presumably it was decided that 'cancer the crab' sounded a lot more sexy than 'cancer the lobster'
I have a separate article on Cancer tattoos.