Just fun note on baby-talk. Scientists use the words child-directed speech for the way in which parents or adults alliterate, rhyme, repeat, and use rhythm and varied pitch when they speak with babies. That is how babies learn to speak and parents that are bi-lingual will have children that are bi-lingual as long as both languages are spoken to the baby. Some families that are let's say Filipino-Americans, for example, will always use English in a formal context and Tagalog at home. The baby will learn both languages this way.
Babbling is specifically an infant's repetition of certain syllables, such as ba-ba-ba, which begins when babies are 6-9 months old. In the first few months of the 2nd year, spoken vocabulary increases at about 1 word per week. When a baby uses a word like Mama or Dada, it's called a "holophrase" -a single word that expresses an entire thought. This is because babies understand about 10 times more words than they can say. Around 21 months is a "naming explosion" in which they know twice as many words. It's called "Naming" because the words they know are usually nouns.
By the age of 5 months, the baby already has a preference for the accents and rhythms of their own culture, so there's not really a global word for the specific area of child language development that you mention. I don't about all cultures but some probably do have a word for it, as you mentioned. The lack of proper pronunciations is simply a process of development and is subjective to the child. All young children master basic grammar according to a schedule, this is called "universal grammar."
Dysfluencies in language – such as stuttering or repeating words or starting sentences over – may be a part of typical speech development as toddlers learn to produce these sounds. A child who may be difficult to understand when they first learn to string words into sentences will usually develop enough articulation over time to be understood. If they don’t, they may have an "articulation delay" or an "articulation disorder."
I hope you might find at least a tidbit of good information in here somewhere.