Forget the 'professional' aspect for a minute - it could all boil down to a personality clash. Maybe your client is used to getting his/her own way and finds it difficult going with other people's ideas - this in turn tries your patience and this comes across to the client.
Lets face it we are not going to get on with every potential customer and it works both ways - but it usually works out as they gain trust in you and vice-versa. In this case it doesnt seem the case.
Although your pride says you must try and win them over, I think you should cut your losses and move on.
Once the deed is done I bet you will feel a sense of relief.
From what you have said, my view is that as a good designer landscaper, you can see through your skills and experience the end result, its always nice taking a friendly passive customer on that journey to an end result ( i did a most simple excavate and returf of 25 sq m of scrub this week and with a nice cutting edge added - the customer was delighted). Its harder when either you are not sure of the 'brief' or they keep changing the initial agreed brief. The reasons they may keep changing may be they don't know what they want (v likely), they don't understand you (unlikely), they are now starting to see what is going to be done and this opens up their own imaginaton (can of worms! and likely) or they are simply a picky pain in the *rse (i think you know the answer here!).
I would do what you have done with design and full itemised quote, covering what HAS to be done (legally), what SHOULD be done (your skills/expertise etc) and thats it - stop there. If more quibble, move on. You wouldn't stop your dentist halfway through a crown and recommend he does it slightly differently. Good luck hope you get your desired result.
Walk Away for definite. If it's bad already it is not going to get any better. We all want to go to work and feel that we are doing a good, satisfying days work - by the sounds of things you are never going to get this feeling with this job.
There are many more enjoyable jobs out there where you will feel far more fulfilled and appreciated.