I think the color confusion starts by using Apple's "ColorSync" to encompass "color management":

• ColorSync this...
• ColorSync that...
• ColorSync calibration...
• ColorSync Profiles...

ColorSync is — merely — a Color Management System (CMS - CMM).
Adobe (ACE), Heidelberg, Agfa, and Kodak CMS/CMMs are also individual Colour Management Systems (CMS), Color Management Modules (CMM).

Further, Apple's "ColorSync Profile" terminology adds to the confusion.
I will argure "ColorSync Profile" is more accurately called "ICC Profile" (in the field of color management).

"ColorSync Profile" = ICC Profile = "ColorSpace"

Here is Apple's ColorSync page at Apple.com.
For more on the ICC International Color Consortium www.color.org.

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Any Color Management System (CMS) — merely — understands ICC profiles and ColorSpaces — and — Converts them to target ICC profiles and ColorSpaces.

The accuracy of the Conversion is soley dependent on the accuracy of the profiles (and the quality of the CMS/CMM software).

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Think of any Color Management System (CMS) as an interpreter of ICC profiles.
The CMS can speak all ICC–profiled languages.

Think of any ICC profile as the written language of each individual, SPECIFIC DEVICE (or ColorSpace) it represents.

Think of your specific monitor ICC profile (MonitorRGB) as an English speaker.

Think of your specific printer/paper/ink ICC profile (TargetICC) as a Spanish speaker.

Think of your Photoshop TaggedFile as a Japanese speaker.

• "TaggedFile" represents the Embedded/Tagged colorspace of the open RGB or CMYK file.
  
  
• If you are Embedding/Tagging your RGB files with "Adobe (1998) RGB" for example:
  
  TaggedFile = AdobeRGB = WorkingRGB

Further: Enabled Monitor ICC Profile = MonitorRGB

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Think of what the Color Management System does:

MonitorRGB < TaggedFile > TargetICC/CMYK

In other words:
TaggedFile is CONVERTED to MonitorRGB.
TaggedFile is CONVERTED to a specific TargetICC or CMYK.

The CMS takes Photoshop's TaggedFile, Converts (translates) it into MonitorRGB (and onto the screen).

The CMS takes Photoshop's TaggedFile, Converts (translates) it into Specific printer/paper/ink ICC (and onto the print).

Without an interpreter, a CMS — our eyes are explaining our color to an English speaker (MonitorRGB) who explains it to a Japanese speaker (TaggedFile) who explains it to a Spanish speaker (Specific printer/paper/ink) — a screen–to–print match is not gonna happen.

See embedded ICC Profiles in action Color Management tutorial
Includes on-line demonstrations of Tagged and UnTagged sRGB, AdobeRGB, AppleRGB
Monitor calibrating and profiling tips, 2.2 gamma, 1.8 Mac gamma, D65 6500 degrees Kelvin

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Adobe Photoshop is a color–managed application.

As a color–managed application:
Photoshop's Color Management System Converts a Tagged File into MonitorRGB (the monitor's "calibrated" colorspace) and thereby displays the tagged file's "true color" through the monitor's "calibrated" colorspace.

In other words, for example:
In a color–managed application like Photoshop — the enabled Color Management System (CMS):

1. CMS Converts TaggedFile to MonitorRGB (and onto your screen).
   
   • If the monitor ICC profile is not accurate, the screen will not be accurate.
   
   
2. CMS Converts TaggedFile to SPECIFIC printer/paper/ink ICC.
   
   • If the printer/paper/ink ICC profile is not accurate, the print will not be accurate.
   

Non–color–managed applications do not "see" the embedded tag, and they do not Convert any color.

• Thus, non–color–managed applications will display a file differently than Photoshop (unless the file was last Saved in the viewing monitor's specific colorspace).

The Basic Theory behind Photoshop Color Management gballard.net.

White Paper about how and when to use Adobe Photoshop Assign Profile and Convert to Profile features:
Includes Photoshop downloads .jpg .tif .png .gif files of Tagged and Un Tagged sRGB, Adobe RGB (1998), Apple RGB and written basic information to understand how Adobe Photoshop handles color profiles.

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PRINTING WORKFLOW:

At the point we enable any printer color management in our Photoshop printing workflow — ColorSync, printer's Automatic, Photo or Sliders — we lose our "straight line," our control of the Conversion: TaggedFile> SPECIFIC printer/paper/ink ICC.

To maintain the strictest control of the MonitorRGB < TaggedFile > PrinterRGB Conversions, see my NoColorAdjustment workflow.

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The Mac OS 9 stores ICC profiles in "ColorSync Profiles" folder, and OS 10 stores them in the Library> ColorSync> Profiles folder, also in the User...Profiles folder.