Here is an explicit move that transforms the given diagram to the standard projection of the torus knot. Consider the upper triangle in the diagram, and label the three crossings at its vertices as 1, 2, 3 from left to right. Now take the (1,2) edge of the triangle and drag this to the right across the vertex 3. The result is the standard projection of a torus knot, where the crossing go "over, over, under, under, over, over, etc." as one proceeds around the knot. (Pictures would describe this isotopy better than words.) I wonder why the tables show the projection they do rather than the one which is more obviously a torus knot.

Here is an explicit move that transforms the diagram shown into the standard projection of the torus knot. Consider the upper triangle in the diagram, and label the three crossings at its vertices as 1, 2, 3 from left to right. Now take the (1,2) edge of the triangle and drag this to the right across the vertex 3. The result is the standard projection of a torus knot, where the crossing go "over, over, under, under, over, over, etc." as one proceeds around the knot. (Pictures would describe this isotopy better than words.)

Added later: Different knot tables seem to show different projections of $8_{19}$. The figure that Ryan added to the original question is the $8_{19}$ knot as shown in what the Knot Atlas webpage calls "The Rolfsen Knot Table", but Rolfsen's book actually has a different projection where it takes two moves of the sort I described to get to the standard projection of a torus knot.