The flaw in your logic is that the edge is partially defined by the vertex your trying to move.

Work arounds:
1. Since you used the split poly tool, you can select the vertex then from the channel box, under inputs, select polysplit. This will show the history for the split operation. From there you can numerically change the position of the vertex along the edge or highlight the affected edge then middle-click and drag in the panel window to move the vertex in 1/10th the edge length increments.

2. Assuming that one end of the split is in the right spot. Use the split poly tool again, anchor to the end that is correct then create a new split to the edge in question. Select the edge of the incorrect split and delete it. Note pressing delete will leave the vertex points created. To delete the edge and it's vertices use the delete edge option from the edit polygons menu.

3. Undo and try again. :-)