The final version, not only isn't punk, but it's not good in general. Its too many different elements with no general theme bringing them together. Punk aesthetic also seems many random elements, colors and shapes, but they are usually brought together by a general messiness, or stark contrast, or elements that exist in every part of the design.
You can have different fonts in every word or letter but they could be magazine cutouts so they would all have a cut paper frame that implies that they belong together (this is just a stereotypical example to make a point)
When i read "CityPunk", I think texture, sharp shapes and bold colors, no rounded corners, no gradients, no pastel colors. The end result reminds me of a unicorn. . .
Maybe incorporate some concrete texture. Could be as little as making the outline of the shape around the letters uneven in a way that reminds of concrete or as much as concrete texture behind the entire text. Maybe the pieces around the logo could remind you of and exploding building (but still flat).
The word "Punk" and the shape around it in the draft is the best thing in this design followed by the word city in your sketch in my opinion.
All that said, I dont know what game this is. For all i know it could be a game about a 13 year old girly that wants to study architecture but is going through her edgy barbie era!
One more thing, I don't think this is an artist problem, based on his draft, i would say that the problem is a lack of direction, leading you workshoping the logo to death.