The vector field of $\begin{bmatrix}\frac{1}{x}\\0\end{bmatrix}$ has a divergence of $-\frac{1}{x^2}$, and -0.25 at the point (2,0) meaning it converges. But at that point, visually, the vector field seems to show more 'output' from the point than 'input'. Isn't convergence the exact opposite of that-- meaning that there is more 'input' than 'output'?
(sorry, don't know how to phrase 'output' and 'input' better)
Your concept of convergence is correct (albeit phrased slightly differently than the usual way). Now if you plot the vector field and recall that divergence is about the limiting value of the flux across the loop enclosing the point $(2,0)$. Then you can tell the 'output' value is less than the 'input' value (people sometimes use the terminology inward or outward flux). See the figure below for reference. Notice that the arrows on the right are shorter than those on the left.