I'm no expert on this, someone else may chime in. I think you have bits of truth from everyone mixed with bits of lore from everyone.
Bricks, if re-using old brick, can be destroyed by using modern mortar. My understanding as to why has to do with the hardness, and therefore the porosity, of both the brick and the mortar. Using hard, new mortar, with old, soft, bricks will cause mosture issues in teh old brick which will lead to spalling. This is especially true in climates with freeze thaw and when they are outside, exposed to the weather.
So, I think they ALL are right to some degree. The last guy can likely get away with regular stuff because its not exposed to weather extremes. The first guy is trying to be most respectful to the original formulation and his approach will work just fine but be costly. The middle guy will possibly come up with something similar to the first guy and his approach will work too because, again, its not exposed to the weather.
Since this is a foundation, inside, unseen - I think you could even consider just re-building it with cinder block, new brick or even concrete and avoid the whole thing about new wineskins and old wine.
I'm not a mason, nor have I played one on TV. I expect a more informed expert to chime in.