How is #H^+# acting in the reaction #H^++:NH_3 ->[H:NH_3]^+#according to the Lewis definition?
H⁺ is acting as a Lewis acid.
A Lewis acid is an electron pair acceptor.
H⁺ ←:NH₃ → NH₄⁺
The Lewis base NH₃ has donated its electron pair to form an N-H bond, and the Lewis acid H⁺ has accepted them.
One-half of the shared electrons in the N-H bond now belongs to the H atom.
You can tell that H⁺ has accepted electrons by counting the formal charges.
Before the reaction, the formal charge on H⁺ is +1. After the reaction, the H has a formal charge of 0.
The H atom has formally gained an electron, so it is an electron acceptor, a Lewis acid.