Minimum spanning tree - Prim's algorithm
Prim's algorithm could give different solutions (depending on how you resolve "ties"), but all solutions should give a MST with the same (minimal) weight, which is not the case here.

Your second solution: $v_0,v_1,v_3,v_2,v_7,v_8,v_6,v_4,v_5$ is incorrect. After adding $v_0,v_1,v_3,v_2$ you add $v_7$. But the minimum distance between $v_7$ and any existing vertex in the tree is $6$. Vertex $v_6$ however has a distance of only $4$, and should thereby be added first. 

You could continue this, and end up with a solution that is indeed different from your solution 1, but has the same weight. For example, $v_0,v_1,v_3,v_2, v_6, v_7,v_8,v_4,v_5$, which gives an MST of weight $37$, as expected.