Lithium niobate on insulator (LNOI) has a great potential for photonic integrated circuits, providing substantial versatility in design of various integrated components. To properly use these components in the implementation of different quantum protocols, photons with different properties are required. In this paper, we theoretically demonstrate a flexible source of correlated photons built on the LNOI waveguide of a special geometry. This source is based on the parametric down-conversion (PDC) process, in which the signal and idler photons are generated at the telecom wavelength and have different spatial profiles and polarizations, but the same group velocities. Such features facilitate electro-optical manipulations with photons and, at the same time, do not require an additional compensation for the time delay. We show how the spectral properties of the generated photons and the number of their frequency modes can be controlled depending on the pump characteristics and the waveguide length. Finally, we discuss a special regime, in which narrowband light with strong frequency correlations is generated at the telecom wavelength.