This is a first attempt at an answer. 

**Source**    
I used your data for X, Y1, and Y2.

    X	Y1	Y2
    1	2	1
    2	6	7
    3	8	9
    4	6	5
    5	10	12
    6	23	18

There is a 1:1 relationship here.  A particular value of X, gives particular values of Y1 and Y2.  The Y values can be thought of as a single point located in a 2d space.  $Y=\left[ y_1,y_2 \right]$

**Procedure**:    

 1. enter the data into excel (excuse any typos)
 2. compute the mean, slope, and intercept using normal methods
 3. compute error between mean and actual for each row
 4. compute error between linear fit and actual for each row
 5. compute sum of squares for the mean-error column
 6. compute sum of squares for the line-error column
 7. compute the ratio of the sums in steps 5 and 6
 8. subtract that value from 1, and compare to the provided R^2

**Results** from approach is shown here:    

![enter image description here][1]

Compute of ratio for RSS shown here:

![enter image description here][2]

Graph of data shown here (yes, y1 label is poorly placed):
![enter image description here][3]


If you have a column of error, and a mean value of the target, then you can compute a Pearson R^2 statistic.

Some relevant references:    

 - https://stats.stackexchange.com/questions/11127/multivariate-multiple-regression-in-r


  [1]: https://i.sstatic.net/8d2U1.png
  [2]: https://i.sstatic.net/9Nsca.png
  [3]: https://i.sstatic.net/41bDj.png