Demonstration of onset of Jahn-Teller (JT) intersections, characterized by a topological phase of pi, on introduction of bending in the collinear tetra-atomic C2H2+ cation, originally a Renner-Teller (RT) system, has raised interest in the study of the generality of this phenomenon. This interest has initiated similar study by shifting one external light atom from the collinear molecular axes of systems such as HCNH and HC2O. Recent studies have revealed that slightly bent HCNH poses a specialty in this regard and thus demands focused attention. In the present work we performed a combined study using both the potential intersections and the angular nonadiabatic coupling terms (NACTs) and report two new results: (i) The first result is the appearance of the JT conical intersections (ci's) between the two lowest states of a slightly bent HCNH molecule (1(2)A' and 1(2)A '', originated from collinear X-2 Pi state) only with certain nonplanar configurations, in contrast to its appearance in C-s configuration space (molecular plane) for slightly bent C2H2+, HC2O, or some other tetra-atomics. This is also associated with the first time demonstration of Berry phase for such a single isolated ci for HCNH molecule. (a) For energetically higher potentials of slightly bent HCNH, the present study reveals the existence of the intersection of the state 1(2)A '' with the state 2(2)A' (originated from the collinear 1(2)Sigma(+) state); in contrast to the (1,2) ci this appears in the molecular plane. The search for ci's has been performed by varying the distances of the two H-atoms (designated as H-C and H-N) from the C-N axis as well as the dihedral angle phi between the two planes (H-C,C,N) and (C,N,H-N). Existence of these JT ci's results in deterioration of a two-state Hilbert subspace (HSS) model in diabatization, while the proper choice of an n (>2)-state HSS circumvents this problem.