Not my invention . . .
Faith is belief. There are all kinds of belief, but only one faith that saves, and that is faith in the righteousness of Jesus Christ.
I do not know what you mean by this term.
I know you didn't personally invent the term, you're repeating what other people invented. I've heard the term before. It's not a scriptural term, it's an invented term. That's my point. If it has validity it must be shown from scripture, not taken for granted. But by the same measure I am free to invent terms if I can establish that it is a reasonable interpretation.
Initial faith? here, I'll demonstrate.
Matthew 13:31-32 KJV
(31) Another parable put he forth unto them, saying,
The kingdom of heaven is like to a grain of mustard seed, which a man took, and sowed in his field:
(32) Which indeed is the least of all seeds: but when it is grown, it is the greatest among herbs, and becometh a tree, so that the birds of the air come and lodge in the branches thereof.
The grain of mustard seed is the initial faith that a man puts forth. It is not "saving faith" but it is responded to and blossoms into saving faith.
Luke 17:5-6 KJV
(5) And the apostles said unto the Lord,
Increase our faith.
(6) And the Lord said,
If ye had faith as a grain of mustard seed, ye might say unto this sycamine tree, Be thou plucked up by the root, and be thou planted in the sea; and it should obey you.
That grain of mustard seed is initial faith. The faith that must be increased. The apostles didn't say they had no faith, but that they needed more faith.
Mark 9:23-24 KJV
(23) Jesus said unto him, If thou canst believe, all things are possible to him that believeth.
(24) And straightway the father of the child cried out, and said with tears, Lord,
I believe; help thou mine unbelief.
Did the father of the child have "saving faith" or did he ask for help towards that saving faith? In this case the response to his faith was evidenced by Jesus's willingness to heal his child.
My point here is that as faith is used in scripture, it's not a binary on / off thing. God requires that we be willing to set aside what we have and be willing to lay down our own life. That is evidence of faith. We are required to put forth something first. God has already demonstrated his intentions through the cross. If we respond to that, if we put forth faith, God can increase our faith. Our faith can be helped by the Holy Spirit to become that "saving faith" you mentioned.
This works in application. It does not require Calvinism. It also agrees with scripture, when Jesus says that people can believe, and then fall away.