You can not "standarize" that much because they refer to diferent things.
Let me explain a bit some units.
Paper sheets are made of, oh well, paper, and the easiest way to measure biiiiiig chunks of paper (Rools actually) is weight.
Weight is the first unit here.
If you make a paper twice the thicness you have twice the material. The first formula is a bit streight.
But as there are diferent sheet sizes: International, American, sometimes with some extra cm here and sometimes extra there, you need have a standard "size" unit, which is a square meter. So the unit is [g/m²] (grammage, gram per square meter, gsm in English)
Lbs or Kg
As a designer I would not "rely" on this unit, because this unit is dependant on the real size of the sheet. If you have a big sheet of paper, 500 pices of this (ream), will weigh more than 500 pices of a small one, both papers of the same thickness and gsm. This unit is for things like transportation.
But in thicker papers, you can not rely on weight alone, but how really thick the cardboard is, so you need a "thickness" unit. Pts. One paper can be lightweight but thick. The unit is a 1/1000 of an inch. 10 pt. is 0.010 in. (This is not the typographic unit, that is a totally diferent one)
You can have some equivalents to pts
You can see them in the table you gave. 3.2 pts = 0.0032 caliper inches. And the mm equivalent is just a matter of dividing them with 2.54.
Type of paper
But the type of paper is diferent, the expected quality of each paper is diferent. Some papers, aditionally to the paper has coatings, which makes the paper to be heavier. Some papers are more dense than others, so given some gsm mean something diferent about how thick and firm the paper is. This depends on the type of fiber, preasure, chemicals, humidity.
But there are diferent brands
One brand could use diferent materials, diferent processes, diferent presure to the press. Each one want to inovate something. So each brand can call its paper No. 1, 2, or any scale they want.
But you are the one
That needs to know the usage, the texture, the sensation. It is not a mathemathical problem.
1) Choose some main types or brands. for example a "Super White Coated" paper.
Buy a sheet of each gsm, and cut them and fold them into a letter size or A4 sample "magazine". I had one where the first page were for example a 100 gsm, the second a 120 gsm, the third 135 etc.
I had one self made catalog of coated paper, one for uncoated, and two for specific types of cardboard.
2) Know the limits of the paper your favorite printhouse has.
Some machines will not print a too thick cardboard because it can potentialy mess up the machine, so it has no sense to have a sample of that paper you will not use.
3) There are thousand types of paper. Some are hand made. I saw one echological that have seeds inside it. If you spend your life making a catalog of them you will not have time to do something with them. So simply Take a decision to work with some brands, and know how they behave.
4) Do not try to be "that creative". I did had some paper catalogs sometime. Unless you are making artesanal stuff, like wedding invitations, there is a chance you will not use them at all.
5) But if you do need to choose exothic papers, do that as a project based decision. Choose one for this wedding. buy it, work with it, enjoy it, and for the next project buy another diferent. Save some phisical samples of papers you know how they behave.
6) Some papers can not really be printed at all. Either too thick, or too much texture, or too absorbent. Use silk print, or some kind of flat bed inkjet as a last resource.