The photon-proton scattering reactions {gamma}p{yields}{pi}{sup o}p,{gamma}p{yields}{gamma}p, and {gamma}p{yields}{gamma}{pi}{sup o}p are investigated in the framework of the three-dimensional time-ordered approach using different models of the on-shell {delta} propagators. These {delta} propagators are taken from the unitary separable models of the {pi}N elastic-scattering amplitude and from the relativistic Breit-Wigner shape representation of the nonunitary {pi}N amplitude in the tree approach. The numerical calculations are performed with the complete set of the one-particle ({pi}-,{omega}-, and {rho}-meson, nucleon, and {delta}) exchange diagrams. It is found that the numerical calculations of the above reactions in the {delta}-resonance region are very sensitive to the model of the {delta} propagator. This sensitivity is explained by the difference between the input {pi}N elastic-scattering observables used for the construction of the {delta} propagator. It is shown, that after additional approximation our calculations reproduce the tree-level model calculations with the Breit-Wigner shape {delta} propagator. This model is in good agreement with the {gamma}p{yields}{pi}{sup o'}p{sup '} reaction and with the preliminary data for the {gamma}p{yields}{gamma}{sup '}{pi}{sup o'}p{sup '} reaction. Conversely, it is demonstrated, that the relativistic Breit-Wigner shape {delta} propagators, obtained from the nonunitary {pi}N amplitudes in the tree approximation, is inconsistent to apply for reproduction of the {gamma}p scattering observablesmore » because they do not describe the {pi}N scattering data in the {delta} resonance region. It is emphasized, that to extract the magnitude of the {delta}{sup +} magnetic moment from the cross sections of the {gamma}p{yields}{gamma}{sup '}{pi}{sup '}N{sup '} reaction it is necessary to fix the form of the {delta} propagator based on the unified description of the two body {pi}p{yields}{pi}N,{gamma}p{yields}{pi}N basic channels together with the essential {gamma}p{yields}{pi}{pi}N and {gamma}p{yields}{gamma}p reactions.« less