To clarify why the behaviour is as it was...
When an intermediate switch is in the up position, L1 and L2 pass straight through from one 2-way to the other, so the switch is transparent. In the down position it crosses over L1 and L2, so that if the two 2-ways match in position it unmatches them, and vice versa. For the 2-ways to function, L1 and L2 'in' must pass through to L1 and L2 'out' either straight or crossed at all times.
With your mis-wired intermediate, in one position the condition was still satisfied that the 'ins' passed through to the 'outs' so the 2-ways functioned normally. If L1 and L2 'in' were on the same side of the switch, that would have been the crossed position. But in the alternative position, the switch spuriously looped each L1 back to the L2 from the same 2-way. (By interchanging the two browns you would have exchanged which intermediate position caused the loop-back)
One of two things would then happen. With 'conventional' 2-way, the older configuration where the permanent live goes to the common of one 2-way and the switched live is taken from the other, the circuit would be broken as there would no longer be any continuity between the two 2-ways. Thus the light would be off regardless of the 2-way positions. OTOH your wiring is 'conversion' 2-way where PL and SL are connected to L1 and L2 at one of the 2-way switches. In this case because the intermediate was looping L1 back to L2, it completed a circuit from PL to SL forcing the light on regardless of the 2-ways.
Therefore, your clearly-worded description:
completely localised the problem to within the intermediate or its connections. Only two possibilities then existed: the switch had an internal short-circuit that only occurred in one position (e.g. because one of its two changeover contacts was operating but the other was jammed) or the wiring was incorrect.
