What is the word in Pali that is translated as being by some and as becoming by others?
The word is "bhava" and in my understanding it refers to a particular state of individualized existence: "being a living being", "being a human", "being a person". So it's not about just the mathematical fact of existence, it is about your subjective position of identifying with this particular organism and its personal story, its narrative.

So when Buddha says "devas and humans enjoy being", and "Dhamma is taught for the cessation of being" - it does not mean, Dhamma advocates death or annihilation, it advocates transcending identification with an individual living being.

Does not mean Dharma advocates identitying with Universe  either, that's another wrong view. Instead, the right view is to see the unhealthy outcomes of any and all identification and to abandon identification altogether.

P.S.

As @ChrisW noted, bhava is also notably mentioned in the Second Noble truth as classification of tanha (craving) into bhava-taṇhā and vibhava-taṇhā. In that context, again, bhava refers to a particular sate of existence - in this case a state of existence that we crave for: "I want to live in a certain way, I want to be X, I want to be surrounded by Y, I want my life to look like Z". Vibhava then refers to not being in a particular state: "I want to stop being poor, I want to stop being sick, I want to stop being lonely" etc.

In all cases bhava seems to always be connected with an idea of self in a broad sense - self not as subject of experience or agent of actions, but rather as one's state of existence.