Classification of time controls in chess

Chess world is moving to faster time controls, probably for the worse. But hey, live and let live. For players who are at beginners or intermediate stage, my advice is to avoid playing bullet, and also avoid blitz. True, blitz can be used to properly train; but 99% players at this stage use it as entertainment or probably think they are practicing, but they are just damaging their chess muscles. Today, I would like to discuss about a different topic: how should we classify time controls into reasonable groups so that there are only a handful of ratings, such as classical, rapid and blitz? This is important because of the introduction of the fast classical time control, and advent of time controls much faster than what could be considered as bullet. 

Conventions in writing (for people new to it):  
Time controls are written as <allotted>+<increment>. For example, 2+1 has allotted time of 2 minutes plus an increment of 1 second per move. In general p+q means p min + q sec. For different units, we write them; e.g.: 10s+0.1 is 10 sec + 0.1 sec.   
Chess.com use Fide's definition in this regard. There is a small difference between chess.com lichess in terms of pre-moves. In chess.com, they cost 0.1 sec whereas they cost 0 sec in lichess (so 10s+0.1 in chess.com is almost the same as 10s+0 in lichess). 

Needless to say, there are arguments as to whether the current classification is reasonable or not, regarding the newer time controls including fast classical and very fast time controls (example). Also, there is no standard classification for time controls faster than bullet. Today, chess.com published a new article reporting that Woodward broke the chess.com bullet rating record. They also mention his last few games before breaking Nakamura's record, and they are hyperbullet games! My first response was "Why should bullet rating change from performance in hyperbullet games?". While preparing to post comment, I noticed a few things. It is clear that Woodward has played some 30s+0 games, and too many 10s+0.1 and 10s+0 games to reach this bullet rating. Personally, I am sure that 30s+0 should be another rating category, and 10s+0 or faster should be yet another rating category. This was obvious. But, then I noticed something more important. The convention of Fide (resp. lichess) to consider the time control p+q under the umbrella of 60 p+ 60 q (resp. 60 p + 40 q) is not really a good idea (details below).  

Classification by Fide or Lichess. 

Official pages: Fide, Lichess
Let us define 'total time' as an estimate of the total time a player gets under a time format, in terms of the estimated number of moves, written here as #moves. Fide uses the value 60 for #moves whereas lichess uses 40. The total time for  p sec + q sec is defined as p + q x #moves (e.g., for 1+1, that is 60 sec + 1 sec, it is 60+60=120 sec for Fide and 60+40=100 sec for lichess). 

The main issue is that with this classification, two different time formats may have the same total time, but they do not belong in the same rating group. For instance, with Fide/lichess convention, 0+15 have the same total time as 1+0 or less! But 0+15 is too fast to be called bullet. The reason for the issue is also easy to notice. The #moves sometimes contribute too much to total time when there is an increment. Hence, I propose the following formula. 

Total time for p sec + q sec time format =  p + p x q + q x 0.5 sec

In the table, 'Total Time' is abbreviated simply as 'time'. Remember that the classification after the new formula is applied is vastly different from using it without the formula. For instance, 2+1 is blitz (whereas it is bullet in lichess). This is for the proper classification. Clearly, lightning is an abomination players should stay away from. 

Thank you for reading.