On behalf of the Name: John did not need to specify whose name this was because early Christians understood that the Name referred to Jesus Christ (see Acts 5:41).
The traveling teachers had chosen to accept nothing offered to them from people who are not Christians. Instead, they relied on the churches for support. Christians who give support to trustworthy traveling teachers share in their work as those teachers spread the truth (see Matthew 10:41–42; Philippians 4:15–19).