When does the brain stem appear in humans?
To correct the article you referenced the neural tube closure normally occurs by the fourth week of pregnancy (28th day after conception). https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3279093/

The Brainstem is comprised of three parts:


Medulla (Melincephalon)
The Pons (Metincephalon)
The Midbrain (Mesencephalon)





  During fetal development
  the brainstem is formed by:
  
  
  the myelencephalon (fifth vesicle) which will develop into the medulla (Melincephalon)
  the metencephalon (fourth vesicle) which will differentiate into the pons and cerebellum
  and the mesencephalon (midbrain)(third vesicle) 
  
  
  An outline of the development of the brain stem is depicted in the image below:
  
  
  
  Pansky, B. 146. Introduction to Brainstem Development. Retrieved from https://discovery.lifemapsc.com/library/review-of-medical-embryology/chapter-146-introduction-to-brainstem-development
  
  
    
    
    By the end of the sixth week the rudimentary development of the five brain vesicles is complete. The cerebral hemispheres have grown and now cover the diencephalon, the mesencephalon and the cerebellum, which has only just begun development.
    
    
    
    Montfort, G. J., &amp; Boon, R. Stages of Brain Development. Retrieved from https://www.bondingandbirth.org/uploads/5/4/1/5/5415260/06_stages_of_brain_development.pdf