Acids have different definitions according to different concepts. Acid is a proton donor according to brønsted lowry concept. According to Lewis concept, acids are electron pair acceptors. Now there are some acids like Boric acid which doesn't donate hydrogen by itself but still its an acid. The reason being it accepts electron pair from OH- and hence obeys Lewis definition. So all acids cannot be explained through brønsted lowry or Arrhenius concept. Here are the three definition of acids:
Arrhenius acids are those which when added to water, increases the concentration of H+ ions in the water.
The Brønsted-Lowry definition is the most widely used definition; unless otherwise specified, acid-base reactions are assumed to involve the transfer of a proton (H+) from an acid to a base.
Lewis acids is a species that accepts a pair of electrons from another species; in other words, it is an electron pair acceptor.