Bhava is very broad, and includes all of “existence”, the current life as well as future lives.
“Craving for bhava” primarily means the wish to be reborn in a future life, whether this is explicitly expressed as an aspiration to be reborn, or is more subtly an “underlying tendency” (anusaya). Either way, the root of the urge is that we are alive now and we want to keep on living. This is why I have sometimes rendered bhava as “continued existence”.
To put the same thing another way: in simple terms, craving for bhava is the wish to be reborn. When reflected upon philosophically, we can understand that that wish comes from our own attachment to this life.
That bhava has primarily a future orientation is made explicit in many places in the suttas, which frequently speak of āyatiṁ punabbhavabhinibbatti, “rebirth in a new state of existence in the future”. If you look closely at the contextual usage, it becomes apparent that this is an expanded form of the more-or-less synonymous bhava.
To specify the current life, a variety of idioms may be used: diṭṭheva dhamme, sandiṭṭhika, akālika, ānantarika, or simply idha. To specify future lives, we have samparāyika, etc.