Part of the problem is, when re-designing, or designing in spec is, you really need to be upfront about the 'worth' of the design. Then there is, how much of the original concept is part of your new design? I'm sure if that was a 'designer' spec, they got paid before they handed the drawing off to get pricing on. Are you going to run into any issue bidding on someone else's work?
On a chance that this will be bid on by other vendors, you could use the example statement you have, and that might cover you for the amount of time you put into the design. But is the design worth more? How much? How will you handle it if they do shop it around?
I think it's better to make sure the client signs off on your copyright statement or when they open the document they are acknowledging that you have copyright of that design and the price of the design fee (whether its a single location, or more for franchise usage) if they choose to take the design somewhere else. Then as an added bonus, you will deduct a portion or all the design at the signature of the construction/installation contract.
Adding a statement with no prior discussion on copyright is not totally covering your butt and may cause problems if they want to take it out to bid... be upfront.