I wouldn't say that first actually "works". It depends on vagaries of NSum implementation, I think. What happens, as in the graphs posted, is that the odd partial sums and even partial sums average to the constant. The sum of Pi terms in such pairs is simply pi (these cancel in the even sums, and cancel with one +pi left over in the odd partial sums). So the contribution of Pi's in the averaged sum below is zero (from -Pi/2-Pi/2+Pi).

((1/2 - Pi/2 + 
     NSum[(-1)^n*(n^(1/n) - Pi), {n, 1, 2000}, 
      NSumTerms -> 2000]) + (1/2 - Pi/2 + 
     NSum[(-1)^n*(n^(1/n) - Pi), {n, 1, 2001}, NSumTerms -> 2001]))/2

If you add those 1/2 terms above you realize that the result, +1, is exactly what would survive had there been 1's in place of the pi's in the two summations. But, had that been the case, we'd be working with the MRB constant by definition.

Upshot: Method 1 only works if there is an averaging done. Which might make it numerically a good method, I don't know.