The answer @Vincent gave is correct in my opinion. I just feel like clarifying a bit (without getting into too many details).
A PANTONE Color Book in Adobe's applications is basically just a list of PANTONE colors with corresponding Lab values. (I know because I once hacked my way through decoding them for a hobby project).
They are stored in .acb files located in your ...\Adobe\Adobe Photoshop CC 2019\Presets\Color Books folder.
The Lab values of the color books comes from making physical prints with the actual physical PANTONE inks and then measuring the resulting color with a measuring device.
No information is giving about exactly which paper has been used or the settings of the printing device, so the measurements are only 100% accurate under the exact same physical conditions as the test print was made under. There are many different paper stocks, printing devices and settings in existence, so the color books are just trying to give us some kind of realistic average (in lack of better words).
The C in the given color codes indicates that you should use the PANTONE+ Solid Coated color book which is based on prints made on coated paper.
When you add a PANTONE spot color swatch in Illustrator or InDesign, you are adding another physical ink to your document. So in addition to the CMYK inks you now have a fifth ink, which must be printed separately (and will increase the cost of printing). You can convert a PANTONE spot color swatch to CMYK or RGB. The color settings of your document will dictate which color profile is used. Beware that many PANTONE colors can't be reproduced satisfyingly with CMYK (or even RGB). That's why they exist in the first place.
When you choose a PANTONE color in Photoshop's color picker and use it to paint on layers or for a solid color layer, it's always converted from the Lab color of the color book to the color mode (RGB or CMYK) and color profile of your document. You can't paint on layers with PANTONE colors!
If you wanted to create a Photoshop document with true PANTONE spot colors, you would have to work with channels, but that's a whole other story.
Your best bet for reproducing PANTONE colors on the web would be to work in sRGB (which is the closest we come to a universal standard) and convert your PANTONE swatches to sRGB. You could also just keep them as PANTONE swatches and wait with the conversion until you export for web.
Since most people don't have color calibrated screens, you'll have to live with the fact that your colors (PANTONE or not) will be displayed differently on different screens.
If you were to make it even more "scientific" you could use a measuring device to measure the actual Lab color of the print on the product, but the color might be outside sRGB gamut and, as mentioned, who knows how it's displayed on all the different screens of the world?