I would like to know is there any way to prove that drawing in a grid on the left would be more precise than in a grid on the right. For me, that make sense since we have more points on the left. But, is there a mathematical way to prove something like this?
If you look at the middle two lines of your left grid, there is a point every $\frac 12$ unit while on the right there is only a point every unit. Once you extend the grid enough beyond the boundary of an area, there will be about twice as many points inside the area with the grid on the left. However, the blocks on the left have the same area of one square unit as the blocks on the right.
You have not defined what style of drawing you are doing or how you define precision. Are you allowed to select line segments from each grid to make the drawing? Do you fill in blocks to color an area? Do you connect dots that are defined by the grid with lines? In the third case, as the left grid has all the points of the right grid and more, you must be able to be more precise. In the first and second, it probably depends on what picture you are trying to draw. If it follows the grid lines on the right, the left will introduce jaggies for vertical lines.