We analyze a sample of 73 old long-period comets (LPCs) (orbital periods $200 < P < 1000$ yr) with perihelion distances $q < 2.5$ au, discovered in the period 1850-2014. We cloned the observed comets and also added fictitious LPCs with perihelia in the Jupiter's zone. We consider both a purely dynamical evolution and a physico-dynamical one with different physical lifetimes. We can fit the computed energy distribution of comets with $q < 1.3$ au to the observed one only within the energy range $0.01 < x < 0.04$ au$^{-1}$ (or periods $125 < P < 1000$ yr), where the "energy" is taken as the inverse of the semimajor axis $a$, namely $x \equiv 1/a$. The best results are obtained for physical lifetimes of about 200-300 revolutions (for a comet with a standard $q = 1$ au). We find that neither a purely dynamical evolution, nor a physico-dynamical one can reproduce the long tail of larger binding energies ($x \gsim 0.04$ au$^{-1}$) that correspond to most Halley-type comets (HTCs) and Jupiter-family comets. We conclude that most HTCs are not the end states of the evolution of LPCs, but come from a different source, a flattened one that we identify with the Centaurs that are scattered to the inner planetary region from the trans-Neptunian belt. These results also show that the boundary between LPCs and HTCs should be located at an energy $x \sim 0.04$ au$^{-1}$ ($P \sim 125$ yr), rather than the conventional classical boundary at $P = 200$ yr.

34 pages, 17 figures, article in press