How would you describe this way of speaking?

Solution 1:

This seems to belong to the family of styles that are termed telegram style, telegraph style, telegraphic style or telegraphese, which is defined in Wikipedia as:

a clipped way of writing that attempts to abbreviate words and pack as much information into the smallest possible number of words or characters.

Example:

Government seized by mystery financier, bubonic plague raging, lovely Spring weather"

A similarly concise style is used in "fridge notes" and resumes. Also, babies use a telegraphic style too, as they're learning to talk. An interesting analysis of what words babies tend to omit or simplify may be found here.