Both definitions are correct. Neither definition is wrong.
The definitions are related, but they refer to different phenomena.
A Brønsted acid is a proton donor. A Brønsted base is a proton acceptor.
A Lewis acid is an electron pair acceptor. A Lewis base is an electron pair donor.
If you want to discuss a reaction in terms of proton transfers, you use the Brønsted definitions.
If you want to discuss a reaction in terms of electron pair transfers, you use the Lewis definitions.