Definitions make it explicit or implicit that to finish something is to complete it, to bring it to a conclusion. For example:
to complete something or come to the end of an activity
Merriam Webster offers many similar examples, all making finish a sequel to start.
Whether the activity was planned or not is irrelevant to its finish. Consider “The car finished skidding”. “The child finished crying”. “The sun finished shining”.
Your example sets out a false consequence by saying “yes”. It implies that you have finished because you had no intention, rather than had finished a previously started washing.
To avoid the false consequence, and if you wish to make a general statement that you no longer intend to wash the car in any circumstances, and that this determines your response to this and any question about washing it, a simpler reply is needed. You may say “I am finished washing the car” or “I am finished with washing the car”. Both statements use finish in the usual defined sense that you used to wash the car but will do so no longer.
This is similar to “I am finished with girls”, which implies that, although the speaker had an interest in girls, they no longer have it.