I have mentioned before. Pantone system is faulty, not perfect, and a lot of times misleading, but it is what we have.
It has evolved because things around have been evolving. Inks, the manufacturing of inks, digital information, ways to measure color, printing methods, printing costs, etc.
The first guides were for spot inks, direct links. In a time where 2 color print was common, where the color selection was not a precise science. The main guide is then the formula guide.
But as now CMYK printing process is now so precise and can be controlled in a good enough accurate way, we need a CMYK based system.
Color Bridge emulates the original Pantone colors to the CMYK workflow, (which is just a way to maintain some hegemony on the choice of colors) In this field, we have different not that well-standardized or well-known alternatives. For example, some "Color Atlas" that are organized in a more logical approach; values of the CMYK inks in an incremental 3dimensional way.
In that regard, the main difference is that the Coated formula guide is a more "absolute" guide. It gives you the spot color, the original color. The Bridge system is therefore not the original color, but an aproximation.
We need to understand a couple of things about CMYK color values.
There is not only one way to produce the same CMYK color. There is, for example, a Chromatic and an achromatic path. Mainly, using CMY colors or using K for those values where the CMY inks neutralize themselves.
Any CMYK value MUST be within a defined CMYK color profile, this is where Pantone guides excel for the lack of clarity.
In my humble tests, for example, the color transformations from a numeric Pantone to CMYK values used on Pantone.com site are using Swop v2 color profile.
This is important. If you live in Europe you would probably be using some Eurocoated or Fogra profile. Japan profile if the case. Even if you are in the U.S. you could use some Gracol profiles. I am used to the SWOP v2 profile. It is old but it still remains a consistent one, especially for the max TAC (Total Area Coverage fo 300%) ink.
In the RGB values, the SWOP standard uses Adobe1998 profile, so that is the one I think the site uses.
My recommendation is that you define your colors as:
Door Number 1
Pantone C, or formula guide as the main color definition.
RGB according to the website (mentioning that it is according to the website and on a specific date)
CMYK values given by the website (also mentioning that)
CMYK values given by the Bridge guide as a backup (also mention that)
Door number 2
Pantone C, or formula guide as the main color definition.
CMYK values according to your current color configuration of your Illustrator or Photoshop (Corel, Affinity or whatever program you are using) but you must specify this profiles software version and ICC profiles and libraries versions...
RGB values exporting the CMYK swatches to RGB with the profiles embedded.
As you can see, door number 2 is more relative, so go with door number 1.
I only would use door number 2, if we are working in a closed environment. If you are the only one using and managing the conversions or it is a small team. But as an identity manual, no. We need more "universal" configurations and values.
Addendum.
If you have the case where you want to choose a color only by watching your screen, then you need to have your screen calibrated with specialized hardware. If you are not, the color calibration of your monitor is not an indispensable issue here.