Well using the scripture Colonel referred to about the devils believe and they tremble, then 'believing with the heart' is not just a mere acknowledgment something is true even if one actually knows it to be true. For instance, the devils
absolutely believe in the existence of God. They
know He exists for a fact. But it doesn't result in salvation or any blessing. But we haven't seen God. So we don't know for a
fact He exists, yet we do. Why? By faith through revelation. But even that in itself doesn't result in salvation. I believed in the existence of God and Jesus because I was raised that way. It was just a 'fact' to me, but I was never saved. So there has to be an actual
trust, actual faith
toward God/Jesus to save us in order to be saved.
Then once we are saved, just being saved doesn't automatically translate into receiving God's other blessings. There is then faith that has to do with receiving what the Bible says is ours. So then we have to
believe in the heart for those things. In Mark 11:23, Jesus didn't say we would receive what we pray for, or receive what we desire, but that we would receive those desires and things we prayed for
if we believe we receive them. So a person can pray or have a desire but not actually
believe they receive that thing. Well we know believing we receive has something to do with speaking and declaring and confession, because Jesus said so in verse 23 and then 'therefored' it with verse 24, calling that action of speaking, etc, 'believe you receive'.
So believing in the heart has to do with believing in an area that is not just the mind. So obviously that would have to do with the spirit, or a combination of the spirit and the mind together (which some say the 'heart' is the operation of the spirit and mind in cooperation with one another), not just the spirit itself.
So anyway, to answer your question, I don't have a definitive answer that I can say 'this is it', but hopefully the answer or at least the path to the answer can be found in what I posted above.