From the example data:
Weighted fitting requires an additional number to
be used as a weight when fitting. The site does
not calculate any weights, which are used as:
error = weight * (predicted - actual)
You must provide any weights you wish to use.
So your data would be entered as:
X Y Weight
5.357 0.376 0.5
5.457 0.489 0.5
5.797 0.874 1.0
etc., etc.
James