Listing Research Universities (Highest research activity) Universities' rankings are misleading, because various institutes publish their ranking results by taking various factors.  

I have found a classification of research universities as R1 , R2 adn R3 here. How reliable is the listing?
The Carnegie Classification ranking system that you link to is fairly reliable. The details of how the rankings are calculated are here and include: 


  
  research &amp; development (R&amp;D) expenditures in science and engineering
  R&amp;D expenditures in non-S&amp;E fields
  S&amp;E research staff (postdoctoral appointees and other non-faculty research staff with doctorates)
  doctoral conferrals in humanities fields, in social science fields, in STEM (science, technology, engineering, and mathematics) fields, and in other fields (e.g., business, education, public policy, social work)
  


They then do a series of statistical manipulations on the numbers to get a final ranking. It seems that the weightings of the factors are less malleable than other rankings (e.g., US News and World Reports). The data seem to be collected from reliable sources (e.g., http://nces.ed.gov/ipeds/). The choice of factors seems malleable and potentially unreliable. It seems like it is likely that if you were to replicate the process with the same factors, you would come up with a similar answer. 

The real question, at least in my mind, is if the factors considered, are at all informative for your needs.