Why is there more than one definition of acid–base behavior?

There is more than one definition because our knowledge of acids and bases increased as the years went by.
In 1777, the French chemist Lavoisier first proposed that acids contained oxygen.
He was wrong. He didn’t know about hydrochloric acid (HCl).
By 1877, the Swedish chemist Svante Arrhenius proposed his definitions of acids and bases.
An acid is a substance that dissociates in water to produce H⁺ ions.
For example, HCl → H⁺ + Cl⁻
A base is a substance that dissociates in water to produce OH- ions.
For example, NaOH → Na⁺ + OH⁻
This meant that acids must contain H atoms, and bases must contain OH⁻ ions.
But there are many well-known bases, such as ammonia (NH₃), that do not contain OH⁻ ions.
In 1923, Johannes Brønsted in Denmark and Thomas Lowry in England independently proposed that acids are proton donors and bases are proton acceptors.
HA + B ⇌ A⁻ + BH⁺
By this definition, NH₃ is a base
HCl + NH₃ → Cl⁻ + NH₄⁺
because the NH₃ accepts a proton from the HCl.
Also in 1923, G.N. Lewis in the United States proposed an alternate theory of acids and bases based on structure and bonding.
His theory involved the transfer of electrons rather than protons.
It stated that an acid is a species that accepts an electron pair, while a base donates an electron pair.
H₃N: + H⁺ → H₃N⁺-H
Thus, NH₃ is a Lewis base because it donates an electron pair to form a bond to the H⁺.