All the information is there but it takes a bit of work.
The DimensionStyle property holds all the information about the LOCAL dimension style applied to the annotation (the chamfer note). This style is saved with the document so even if it is modified, it will be the style referenced by all annotations that use that style
Modifications to a chamfer note (also applies to other annotations) are stored with the annotation itself. In the case of a standard Inventor chamfer note, this is done in the UI by editing the chamfer note, clicking the Precision and Tolerance button and then unchecking the Use Global Precision checkbox. The user can then override the precision and tolerance values for the included parameters, Distance1 and Angle for the standard chamfer note.
To do this programmatically you just need to change the FormattedChamferNote property to include the specific overrides. As an example, the following two values for a standard chamfer note and a user overridden (via the UI) chamfer note for the same annotation show the differences where the precision of the Distance1 value has been changed from 2 to 3 decimal places, and a symmetrical tolerance of .005" (0.0127cm) was added.
ORIGINAL CH NOTE
<Distance1 Precision='2' AlternatePrecision='2' UpperTolerance='0.000000' LowerTolerance='0.000000' ToleranceType='kSymmetricTolerance' TolerancePrecision='2' ToleranceAlternatePrecision='2'></Distance1> X <Angle Precision='2' UpperTolerance='0.000000' LowerTolerance='0.000000' ToleranceType='kSymmetricTolerance' TolerancePrecision='2'></Angle> Chamfer
CH NOTE FOR MODIFIED CHAMFER NOTE
<Distance1 Precision='3' AlternatePrecision='2' UpperTolerance='0.012700' LowerTolerance='0.000000' ToleranceType='kSymmetricTolerance' TolerancePrecision='3' ToleranceAlternatePrecision='2'></Distance1> X <Angle Precision='2' UpperTolerance='0.000000' LowerTolerance='0.000000' ToleranceType='kSymmetricTolerance' TolerancePrecision='2'></Angle> Chamfer
The DimensionStyle.LinearPrecison property has a value of LinearPrecionEnum.kTwoDecimalPlacesLinearPrecision, which represents two decimal places. This information is read-only and does not change when the user overrides the chamfer note. Styles can recognize these enumerated values but the FormattedChamferNote requires the precision to be character(s) that represent the numerical value (e.g. 3). You would have to convert the enumerated value in the Dimension Style to a numeric value to compare it to the current numerical value (from the string) in the FormattedChamferNote.
If your code needs to do this on a regular basis, consider writing smaller functions that do singular tasks, such as
Function IsChamferPrecisionOverriden(chNote as ChamferNote) As Boolean. You would compare the appropriate dim style property(ies) and value(s) extracted from the FormattedChamferNote to see if the are the same, and return the appropriate true or false value.
So it will take some work but that is often the case. There are plenty of cases where the Inventor API does not quite work the way it may be implied in the help file. Best of luck with this.