"faul" is pretty much a 100% translation of lazy. It does not have a separate procrastinatory meaning.
That itThe proverb refers to lazy peoples' habit of promising to do something "tomorrow" even though in the end, they mightnever do it. Even though the same words might be used by a very busy person who is not lazy is a shortcoming of the proverb.too, :)faul relates to laziness only.