The exact and approximate hadron transverse momentum distributions for the Fermi-Dirac, Bose-Einstein and Maxwell-Boltzmann statistics of particles in the framework of the Tsallis statistics with escort probabilities (the Tsallis-3 statistics) have been derived. The classical and quantum transverse momentum distributions in the zeroth term approximation and the quantum transverse momentum distributions in the factorization approximation introduced in the zeroth term approximation were found. The transverse momentum distributions in the zeroth term approximation and in the factorization approximation of the zeroth term approximation are the same in the Tsallis-3, Tsallis-2 and q-dual statistics. The well-known classical phenomenological Tsallis distribution exactly coincides with the classical transverse momentum distribution of the Tsallis-3 statistics in the zeroth term approximation for which the entropy of system is zero in the whole range of state variables. However, the quantum phenomenological Tsallis distribution does not coincide with either the exact or approximate transverse momentum distributions of the Tsallis-3 statistics. The exact Tsallis-3 classical distribution and the classical phenomenological Tsallis distribution were applied to describe the experimental spectra of the charged pions produced in the proton-proton collisions at high energies. The values of the parameters (T, q) for both these model distributions differ in the whole energy range. Thus, the classical phenomenological Tsallis distribution is an unsatisfactory approximation for the exact classical transverse momentum distribution of the Tsallis-3 statistics.