Different educationist categories it. In one school of thought education start when child born. Reading and writing can be named as early childhood reading and writing or ECRW
I have a very broad perspective on what counts as literacy so early literacy works fine. I think it is more common in the literature these days and emergent.
1/ I worked 50 years ago in this field, which at the time was highly topical and related to extensive empirical research and the development of new curricula. Some texts and studies can be found in my research section on RG.
2/ As far as I can see, there are no major movements in preschool education today, and you can't help but work your way through a variety of textbooks.
3/ The overview is made difficult by the fact that pre-school education in the individual countries - especially in Europe and the USA - has very different traditions, which date back to the time of classical reform pedagogy, i.e. are about 100 years old today.
4/ The educational race between the USA and the Soviet Union after the Second World War was strongly extended to pre-school education, after the Soviets had the Sputnik orbit the world in autumn 1957. America attempted to compensate for the educational deficit of the public school system, which was attributed to John Dewey's liberal education doctrine and its broad supporters: through the Head-Start Programme, which with some delay also led to new educational reforms in Western Europe, above all in Germany. From this time on, it is worthwhile to follow the further
I am just an undergraduate student with former and ongoing professional activitiy in ECE, so take my words with a pinch of salt.
I have always thought that "emergent" is a "politically charged" adjective, used by educators and researchers who base their intervention on the core ideas of children as active learners and of learning (and curriculum) emerging from the children's interest through the adult's attitude towards active listening and through democratic processes and interactions between adults and children, with relationships of power kept in mind and and under control. I guess that much of this comes from Reggio Emilia, at least in my limited experience and knowledge.
So, "emergent literacy" is literacy-related learning that arises from these processes in the first years of a child's life, since birth. There is a focus on children as complete human beings in the here and now, although in development.
On the other hand, "early literacy" reminds me of a broader definition of literacy for very young children, charged with different values. I think of the whole concept of early intervention with its focus on outcomes: think of what Mr. Retter wrote above about American programs for early intervention in disadvantaged areas.
I have never thought of the two definitions as mutually exclusive, but sure they point at nuanced differences in our approach to pedagogy, or maybe simply in our focus of the moment.