I wanted to know if the legends of vampires being vulnerable to a crucifix or a cross exist because of what happened to Jesus. Is this so?
This question might be of interest.
If it's correct that modern vampire folklore largely derives from Stoker, then most likely the answer to your question is yes. Vampire literature only seems to predate Dracula by a century and a half, so it doesn't appear that we're really looking at any kind of deep lore here.
Whether in legend or real life, the crucifix is a very powerful sign indeed.