Best Practice for colours that are different between applications

I am trying to create a colour swatch for a brand based on the branding guideline supplied by the original designer.

In their branding guideline they have supplied the following colours (written out as supplied by them):

PMS Proc Blue - c100m10y0k0

PMS 369 - c60m0y100k0

black (no CMYK supplied)

white (no CMYK supplied)

File exported as .png directly from Illustrator:

So I loaded these values into Adobe Illustrator and this is what I got:

c100m10y0k0 - #00a1e4 - r0g161b228 (PMS Proc Blue based on CMYK values supplied)

c77m29y0k0 - #0094d8 - r0g148b216 (PMS Proc Blue from the filled shape in the file sent by designer - opened in Illustrator)

c60m0y100k0 - #72bf44 - r114g191b68 (PMS 369 based on CMYK values supplied)

c58m7y100k0 - #7ab51d - r122g181b29 (PMS 369 from the filled shape in the file sent by designer - opened in Illustrator)

c73m70y62k78 - #1a171b - r26g23b27 (black from the file sent by designer - opened in Illustrator)

c0m0y0k100 - #231f20 - r35g31b32 (black based on default "print" swatch library CMYK values in Illustrator)

c0m0y0k0 - #ffffff - r255g255b255 (white based on default "print" swatch library CMYK values in Illustrator and the filled shape in the file sent by designer - opened in Illustrator)

Then I loaded them into color.adobe.com and this is what I got:

c100m10y0k0 - #00E5FF - r0g229b255 (PMS Proc Blue based on CMYK values supplied)

c60m0y100k0 - #66FF00 - r102g255b0 (PMS 369 based on CMYK values supplied)

c0m0y0k100 - #000000 - r35g31b32 (black based on default "print" swatch library CMYK values in Illustrator)

c0m0y0k0 - #ffffff - r255g255b255 (white based on default "print" swatch library CMYK values in Illustrator)

So I decided to check the Pantone website (using Find a Pantone Color | Quick Online Color Tool and searching by the CMYK values provided), and this is what I got:

c100m12y0k2 - #008DCE - r0g141b206 (PANTONE 2394 CP) (Blue)

c100m13y1k2 - #008BCC - r0g139b204 (Alternative 1 - PANTONE Process Blue CP. So at least I found one close to the name supplied with the CMYK, but still not bang on)

c59m0y100k0 - #6DAC4F - r109g172b79 (PANTONE 3501 UP) (Green)

c0m0y0k100 - #544F4B - r84g79b75 (PANTONE P 179-16 U) (Black)

c0m2y0k0 - #FCF6F5 - r252g246b245 (PANTONE P 75-1 U) (White)

Now I'm really confused as to what the actual colour values are!

I'd be happy to use the values provided from the Pantone site (however I'm not sure about what values I should use for the black and white), but when I put those values in Illustrator and color.adobe.com, they still don't match each other. All 3 sources output differet values.

Can anyone advise me on what would be best practice here? I've never come across this issue before. I'm a web designer by trade so I've always worked off the hex values, but I want them to be correct to the design...