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In your method 2: if you say that you regress the portfolio return $r = \sum w_i r_i$ on the asset returns $r_i$ then you do multivariate regression and all covariances between the assets will be incorporated in the solution (the vector $\beta$). Using method 1 then you first calculate univariate regressions and weigt them - this is something different.
Mathematically they must be the same: $\frac{Cov(Portfolio_{returns},r^m)}{Var(r^m)} = \frac{Cov(\sum w_i r_i,r^m)}{{Var(r^m)}} = \frac{\sum w_i Cov(r_i,r_m)}{Var(r^m)} = \sum w_i \beta_i = Portfolio_{beta}$ This is just math and has nothing to do with finance. They must yield the same.