I have been under the impression that Way of the Crab required the friendly crab personality to leave play to take effect. This would mean that if you used an effect to prevent the personality from leaving play, your opponent would not have to lose their personality. Is this the case or have I been playing it wrong?

Rulings found here

https://fiveringsdb.com/card/way-of-the-crab

Sacrificing a character for Way of the Crab is part of the cost - if the costs are not paid, the event is not played. Therefore - if you prevent a character from being sacrificed then you are not paying the cost.

Rules Reference page 4-5 - Costs bullet 4

Also they cover this explicitly in the FAQ at the end of the Rules Reference.