Believing of any kind, if it's more than merely superficial, intellectual assent, is inevitably reflected or manifested in corresponding action. If I really believe wearing a life jacket when on the water will be important in keeping me safe, I'll put it on, even if it's hot and uncomfortable. If I really believe my dentist can fix my painful tooth, I'll go to him to get it fixed, even if doing so is unpleasant and expensive. If I really believe my mechanic can fix the wobbling front tire on my car, I'll take my car to him to repair, even though it's inconvenient and costly to do so. And so on. It isn't, then, ONLY concerning spiritual matters that belief and action correspond.
Such corresponding action, though, merely completes belief, it doesn't constitute it. If, for example, I believe the snarling dog running at me is going to bite me, I may not have time to act on my sincere and certain belief and climb a tree. Does this mean my belief didn't exist, or wasn't genuine and sincere? No. This example illustrates that belief always precedes action and exists discretely separate from the action it may produce. In other words, I can truly believe something and yet not manifest that belief in concrete action. The man on his deathbed, immobilized by cancer, who trusts in Christ only moments before he dies, is truly saved though his belief has no opportunity to reflect itself in the dying man's conduct. This is so because his deeds, his God-obedient actions, aren't what saves him (Titus 3:5; Ephesians 2:8-9; 2 Timothy 1:9). Salvation is a Person, Jesus Christ. He alone saves. (Acts 4:12; John 14:6; 1 Timothy 2:5)
Only insofar as they often overlap with each other. But as the examples above demonstrate, they aren't actually identical things.
This simply illustrates the inter-relationship of belief with action, not that they are synonymous. What did Jesus say about belief in him and the work of God?
John 6:28-29
28 Then they said to him, “What must we do, to be doing the works of God?”
29 Jesus answered them, “This is the work of God, that you believe in him whom he has sent.”
Trusting in, believing in, Jesus as Saviour and submitting to him as Lord (Romans 10:9-10) are things we do, though they are not doings of the external-action sort, like leaping up into a tree when a snarling dog charges at you. They are entirely of the heart, mind and will, internal and unseen by all, save God.
The dentist doesn't tell us to do our own root canal upon ourselves. That's silly. If we could do such a thing, we wouldn't be at the dentist's office! We see the dentist because we want him to fix our tooth. Though our belief in our dentist has resulted in our going to him for help, in the end, we can only receive from him his tooth-repairing work. We don't fix our own tooth, just as we don't save ourselves.
"There is no man so blind as he who will not see." Nothing in this quotation from your post does anything to counter or negate my point about the limits of our belief.
Yes, but the Source of all Truth is not identical to the truth propositions that arise from him.