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No, paying your ticket very much does change the legal process. The government may be free to seek other sanctions in court if you go there, and by paying up and giving in to the first demand, you preclude them from that opportunity. You change the legal process by giving up the money and not contesting the fine.

Every action you take in a legal process is a "change" of that legal process. At issue here is whether the plaintiff/claimant can demand a jury trial. Juries decide facts, and judges decide the law. By giving up the money, Google is effectively saying they won't be contesting the facts of the case. That means no need for a jury.