We need to be careful to consider the auxiliary coordinate variable case as 
well as the dimension coordinate variable case. It's quite true that one 
person's coordinate may be another person's data. Time is a good example of 
this. I have done work where I did analysis of the contents of the time 
variable to understand something going on in a dataset. This is clearly less 
true of grid X or Y variables (or Lat or Lon) in the case where the values are 
static. Auxiliary coordinate variables often exist in a gray area between 
static coordinate and "pure data".

How about one of these two sets of paragraphs? One set excludes "string x(x)" 
as a valid dimension coordinate variable, and the other allows it. Both exclude 
auxiliary coordinate variables of the form "x(x)".

> Coordinate variables contain values that locate one or more sets of data 
> values with respect to a set of axes of a (possibly abstract) coordinate 
> system. Each axis of the coordinate system may be temporal, spatial or 
> represent some state of the system such as energy level or mode. There are 
> two types of coordinate variables — dimension coordinate variables and 
> auxiliary coordinate variables.
>
> A dimension coordinate variable shall be a one-dimensional variable that has 
> the same name as its dimension. A dimension coordinate variable shall have a 
> numeric type and the contents shall be strictly monotonic. A dimension 
> coordinate variable shall not contain fill, missing, or out-of range values. 
> A one-dimensional variable that is not a dimension coordinate variable shall 
> not have the same name as its dimension.
>
> An auxiliary coordinate variable is an N-dimensional variable (where N is 
> greater than or equal to zero) that contains coordinate values that cannot be 
> expressed in the form of a dimension coordinate variable. A two-dimensional 
> variable that contains latitude values for the cell centers of a 
> two-dimensional spatial grid that is not aligned to fixed latitude and 
> longitude axes is an example of an auxiliary coordinate variable. Auxiliary 
> coordinate variables may have non-numeric type and the contents are not 
> required to be monotonic or unique. Auxiliary coordinate variables may 
> contain fill, missing, and out-of-range values. A one-dimensional auxiliary 
> coordinate variable shall not have the same name as its dimension.

or 

> Coordinate variables contain values that locate one or more sets of data 
> values with respect to a set of axes of a (possibly abstract) coordinate 
> system. Each axis of the coordinate system may be temporal, spatial or 
> represent some state of the system such as energy level or mode. There are 
> two types of coordinate variables — dimension coordinate variables and 
> auxiliary coordinate variables.
>
> A dimension coordinate variable shall be a one-dimensional variable that has 
> the same name as its dimension. A dimension coordinate variable shall have a 
> numeric type and the contents shall be strictly monotonic, or it shall have 
> string type and each value of the contents shall be unique. A dimension 
> coordinate variable shall not contain fill, missing, or out-of range values. 
> A one-dimensional variable that is not a dimension coordinate variable shall 
> not have the same name as its dimension.
>
> An auxiliary coordinate variable is an N-dimensional variable (where N is 
> greater than or equal to zero) that contains coordinate values that cannot be 
> expressed in the form of a dimension coordinate variable. A two-dimensional 
> variable that contains latitude values for the cell centers of a 
> two-dimensional spatial grid that is not aligned to fixed latitude and 
> longitude axes is an example of an auxiliary coordinate variable. Auxiliary 
> coordinate variables may have non-numeric type and the contents are not 
> required to be monotonic or unique. Auxiliary coordinate variables may 
> contain fill, missing, and out-of-range values. A one-dimensional auxiliary 
> coordinate variable shall not have the same name as its dimension.

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