The basic meaning of bhava is “existence” or “life”. “Becoming” is an incorrect translation, as bhava is a countable noun, eg. a stream-enterer “does not take an eighth existence”, i.e. they are only reborn seven times at most.
In its straightforward usage, bhava means “existence” or “life” in the sense of “in my past life I was Cleopatra”. But it has a more philosophical sense: we exist now, and will continue to exist in the future. So there is a continuity of “existence” which manifests as particular “lives”.
“Craving for existence” is not “craving to become”, it means “craving to continue, to survive, to be”, especially, the craving to be reborn in future lives. It can be a conscious wish or an unconscious attachment.
Modern interpreters often overlook the close connection between bhava and rebirth, so I chose to emphasize this connection in my translations.