这都要根据语境来决定：
the better (of the two) 用来表达“两个"之间比较好的一个,正如 the best 用以表达两个以上中 ”最好“ 的一个.
Of the two players, I think John is the better one.(the better of the TWO)
I think LeBron James is "the best" player In the entire NBA.

a better 的用法, 后面必须跟一个可数名词：

She is definitely a better "singer" than XXX.跟XXX比,她绝对是更好的（”一个“）歌手.