What you see in those text books is correct. I don't know if your confusion stems from the older definition of NH3 or not but here it is for what it's worth. In the older days (when I was in school), it was thought that NH3 gas dissolved in water to form NH4OH as follows:
NH3 + HOH ==> NH4OH.
Then, NH4OH ionized (partially since it is a weak base) to form the following:
NH4OH <==> NH4^+ + OH^- and we went with that thought for many many years. It has been shown, however, (and I don't know the exact date but I think it was sometime in the 1960s or 1970s) that NH4OH does NOT exist. Therefore, ammonia may be described as a base by both the modified Arrhenius definition as well as the Bronsted-Lowry theory.
By the modified Arrhenius we say a substance is a base if it produces OH- in solution. NH3 does this by
NH3 + HOH ==> NH4^+ + OH^-.
By the Bronsted-Lowry theory, a base is a proton acceptor. NH3, remember, has a pair of unshared electrons, shown here as :NH3 + HOH ==> NH4^+ + OH^-. Note that the :NH3 has accepted a proton (from water); thus, NH3 is a base and water is an acid. The reaction forms a conjugate acid (NH4^+) and a conjugate base (OH^-). I still see some reference now and then about NH4OH and it's ionization into NH4^+ and OH^- but the part about the existence of NH4OH is not true.