Both statements are correct. Only left-handed electrons and left-handed neutrinos participate in weak interactions. The projection operators $$ P_L = \frac{1}{2}(1-\gamma^5)\\ P_R = \frac{1}{2}(1+\gamma^5)\\ $$ satisfy the relations $$ P_L \gamma_\mu = \gamma_\mu P_R\\ P_LP_R=0\\ P_LP_L=1\\ P_L + P_R =1 $$$$ P_L \gamma_\mu = \gamma_\mu P_R\\ P_LP_R=0\\ P_LP_L=P_L\\ P_L + P_R =1 $$ From this it follows that $$ j^\mu=\bar u_e \gamma^\mu P_L u_\nu\\ = \bar u_e (P_L + P_R) \gamma^\mu P_L u_\nu\\ = \bar u_e \gamma^\mu (P_R + P_L) P_L u_\nu\\ = \bar u_e \gamma^\mu P_L P_L u_\nu\\ = \bar u_e P_R \gamma^\mu P_L u_\nu\\ = \bar u_{Le} \gamma^\mu u_{L\nu} $$$$ j^\mu=\bar u_e \gamma^\mu P_L u_\nu\\ = \bar u_e \gamma^\mu P_L P_L u_\nu\\ = \bar u_e P_R \gamma^\mu P_L u_\nu\\ = \bar u_{Le} \gamma^\mu u_{L\nu} $$ where in the final line I use the fact that $$ \bar u_{Le} = \overline{(P_L u_e)} = u_e^\dagger P_L^\dagger \gamma_0 = \bar u_e P_R $$
