You get a round dot with a green because the diode has really nothing to do with the beam you get out. The diode just provides energy for the Nd:YVO4 to start lasing (you could also use a flash lamp, arc lamp, etc... it just needs high density light energy around 808nm). From there, the Nd:YVO4 is creating it's own beam, completely separate from the 808 pump light. They generally lase tem00, because that's what the cavity most easily supports, which is why you get a nice round beam
There is much more science behind it to explain exactly WHY the crystal usually wants to lase tem00, but it goes up and beyond my level of knowledge.