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This looks fine. The agreement states the exact amount to be paid, right?
Then you should be all set. Let me know if you need anything else.
It is very unlikely that anything would ever come of it, but if you use the landlord's suggested language you could potentially be liable for something that happened while you were living there if it wasn't expressly contemplated in the lease agreement. That said, I'm having a hard time coming up with an example.
It seems odd that the landlord would care this much. If you hold firm, he might cave in. On the other hand, if there has been no other damage or any other issues with the apartment, you should be fine signing his version.