Indesign - CMKY conversion

The LAB values of the Pantones are measured directly from a swatch of actual Pantone Ink on a printed sheet. In a colour managed system, this is the value you WANT to match, as the point is to match Pantone INK. LAB is used because it's a very big gamut - pretty much every colour we can see as humans. RGB is a smaller subset of that, and CMYK, of course, is even smaller. Pantone inks fall within the LAB space but may be outside CMYK and RGB.
Pantone matches in desktop publishing applications since the "olden days" have changed with the advances in the industry. e.g. In the early days when CMYK printing was pretty much solely offset printing related, Pantone converted their colours into a CMYK mix that would work "generally" in that world (the old CV, CVC values etc). As such, the Pantone spot colours in older applications were defined with CMYK internal definitions. Nowadays, with the many many different ways of printing full colour work (e.g. inkjet, dye sublimation, etc etc) the CMYK mixes that would work on an offset press would print completely differently using the much more vibrant CMYK (or more) inks of some of these new printers, and in fact, these wider gamut printers could print more Pantone colours accurately than before. Hence, Color Management became necessary to bridge the trip from what colour you are trying to match to what the final printer can actually accomplish.
Pantone's Color Bridge was an improvement of the earlier days' CMYK-based definitions, as offset printing processes are much improved now with things like Computer to Plate technologies, where there is more accurate reproduction of screen values, with less gain, etc etc. Still, the CMYK values in Color Bridge are Pantone's definitions on what is the closest to a match of their spot inks, albeit still in a very specific print workflow, but much more typical of an offset print workflow you'd see nowadays. So if you need to spec CMYK values, say in a brand guideline manual, you should use these. (I can't speak to Pantone Connect and how they are determining CMYK values).
What you should NOT do is spec CMYK values you get from your own specific system; When you convert a Pantone LAB value to CMYK in your application, say InDesign, you are getting values based on YOUR specific CM situation, and may be completely incorrect for the actual final printer, and also, will probably be quite different than what a colleague might get in their system if they have different CM settings from you. For example, say your CMYK setting in InDesign is SWOP Sheetfed Coated (which is a very old profile based on film and plate technology that had considerable gain), but you are printing to a modern press that might be matching FOGRA or GRACOL with very little gain, your colours will print slightly lighter/washed out.
So often, I see brand guidelines from clients from supposedly reputable ad agencies where they have done similar LAB to CMYK conversions, and have wondered why noone else can get the colour right.
The other thing to consider is that some modern RIPs have Pantone Replacement, where if it sees a spot Pantone in a PDF file, it can replace it with a process definition that matches better on the specific press, and discard your CMYK definition.

A lot of this depends on the printer/rip for the printer. Spot colors are mainly useful in how the printer/rip can alter colors. If they have the ability to adjust colors there, the easiest way is with a spot color. If you are having to make color adjustments before getting to the printer/rip, then spot/process color does not matter as much although having them as global colors will make color adjustments easier.

That being said all printers and rips have the ability to interpret colors in different color spaces, but that does not mean they are all equal. Depending on the color, the printer, the rip, and the profile linearization certain colors will look better. I agree with Scott Falkner on using a process swatch book to see what pantone says is the closest process color, but the color space that you use should be determined more by the process and capabilities of the printer being used.

If I am printing process colours i will define all my swatches in CMYK as process colours. If I have to match a solid colour in process (such as a PANTONE) I usually don’t rely on the Bridge or the process equivalent swatch book. I have both solid and process swatch books so I will visually match the colour as best I can.