The whole region was called Palestine, as you explained. Including Jews, Arabs, everybody. In today's environment, there's a specific subset of people calling themselves Palestinians, claiming that they are the rightful owners of a specific land and country called Palestine.
The Mandate was a legal instrument allowing Britain to rule/administer the region. As far as I know, no self-governing authority ever took power. And if there's been no self-governing authority or government in power, I don't see how it can be called an actual country. Especially since there were plans from the start to divide the territory into smaller entities.
According to wikipedia, "The mandate document formalised the division of the British protectorates - Palestine, to include a national home for the Jewish people, under direct British rule, and Transjordan, an Emirate governed semi-autonomously from Britain under the rule of the Hashemite family."
There's also no mention there of any third division of the land for the rocket scientists who ignited the present crisis.