The only point we seem to disagree on is:
"the mind is a subproduct, lies inside, or depends on the soul"
I would say that they are simply different aspects of the same thing.
This is going to get a bit esoteric, so bear with me. The average Japanese
person views the soul as something with substance/essence, but not on the
physical plane -- a non-physical substance, per se. (And if you can
rationalize that, you can also understand the sound of one hand clapping.
)
This is why in Japanese anime and fairy tales, spirits/souls are often shown as
an ethereal flame, flickering light or other apparition. However, this is
merely a non-physical manifestation of the true essence of the soul, which is
best described as the "will" (heart/mind/intellect/intention/etc.).
Buddhism views the soul more like Western religions, but in the original
Japanese religion (Shinto = animism), it is the "will" (= ishi) that matters.
For example, when an inanimate object (tree, rock, waterfall, sword, etc.) is
said to have a "soul", this can happen in a couple of ways: 1) The object can
develop a "will" on its own, or 2) some other "will" can take up residence in
the object. For example, a sword can take on the will/intention (pattern) of
its owner (and sometimes even its victims), or upon its death the will of a
faithful dog transfers to a nearby object so that it can continue to watch over
and protect its master, and so on. In this case it is the "will" (heart/mind)
that is the initiating factor, and the soul is the manifestation or byproduct,
although this distinction is not usually made in Japan.
However, in the West, the soul is the prime aspect, and as you say "the mind is
a subproduct". This creates a problem when translating not just the words but
also the underlying meaning, so since the distinction is not so great on the
Japanese side, it is more correct to use the term "soul" in cases like Eva
because that is what the audience will understand better and which more
accurately conveys all of the nuances involved. IMO "mind" is too sterile
because it implies simple brian patterns which is against the premise of Eva.
"Heart" would be okay, I believe that "soul" is more comprehensive in terms of
equivalent English nuances.
, both the subs and dubs of DEATH say "Lilin". 
