I was wondering if anyone knows if you use the Ink Manager to convert Pantone Spot Colors to CMYK in InDesign is the outcome an official Pantone recommended value?
Short answer: No.
They might be close, but will almost certainly not match exactly. Adobe's Pantone swatches libraries are based on Lab gamut values, which do not convert directly to CMYK gamut values. If you convert a PANTONE+ Solid Coated swatch for PMS 300 (L:36 a:-10 b:-32) to CMYK for example, the result is C:100 M:62.21 Y:7.03 K:0.37. The Pantone color bridge that I have for solid coated shows C:100 M:44 Y:0 K:0. There may be some instances where the Lab values translate to the same CMYK values listed in the Pantone bridge guide, but it will be by chance.
Note that these values were from the InDesign CC2018 Pantone libraries, which may differ between CC releases and applications like Illustrator and Photoshop. Also, InDesign's Pantone+ Coated Bridge swatch values for PMS 300 (C:100 M:50 Y:0 K:2) do not match my printed Pantone Bridge guide, so they don't even match each other from year to year.
Initially, yes, because big brands work closely with each other. Adobe, Corel, and QuarkXpress work with Pantone to have the latest values when software is released. So the libraries used are provided by Pantone.
But no, because there is not one conversion, there are many. A Pantone to CMYK conversion is made by a matrix of values taking into account the Destination Color Profile.
A problem is that Pantone is changing the values of these conversions from time to time. Another is that they do not explain to people that some "official values" are specifics to one profile. The ones on the website, as I tested them some time ago, were based on a Swop v2 Profile, and they are not the same as if you use some Fogra Profile on your application.
Some time ago you could download the libraries for free. I think you can do that with the purchase of some Pantone guides, or pay some USD.
That is probably the best way to have the latest "Official Pantone Values".
But personally, I would NOT update the Pantone values until the next release of the software you are using. If you update the libraries, some recent project you did with a specific library can potentially change values after you update. This is not a good scenario.
After a new release, it is expected to have some changes.