[Smaug123]

A sentence which fails to specify a real uniquely… fails to specify a real uniquely. Borel's talking about numbers which can be defined uniquely: that is, picked out, identified. Your Berry-paradox description doesn't identify an integer.

[zodiac]

But if we assume that we can say a sentence either specifies a real number uniquely or does not, the Berry paradox number is uniquely specified.