Answer

Your question has two parts, which I'll answer in order.

1) As used in theological discourse, "impute" is synonymous with verbs like "account," "count," "reckon," "credit," "accredit," all of which are used more or less interchangeably. A difference in meaning should not be sought between "impute" and "reckon."

As to the comment, "This word deals with reality, etc.," that is true, but only in a legal sense as it concerns justification (God's just judgment that someone is righteous), not in the sense that someone's personal disposition and conduct is righteous. Remember, the statement about imputation/reckoning (Gen. 15:6) is quoted precisely as Abraham is contemplated as "ungodly," not righteous in his action (Rom. 4:5).

God's justifying judgment about a person is true, that is it "deals with reality" because of Christ, specifically because, by God's design, Christ is that person's representative and substitute so that when united to Christ by faith, Christ's righteousness is reckoned or imputed to that person, even though that person remains unrighteous (sinful) in disposition and conduct. This is at the heart of the gospel. Justified sinners, inseparably from their being justified, are also being sanctified; that is, alive in Christ, they are being renewed in righteousness and holiness in their thoughts and actions. But their sanctification (neither their faith nor the obedience that flows from faith) does not come into consideration for their justification (see the statement from the Westminster Confession of Faith below).

2) To understand Romans 4:3, 5 (see also Gal. 3:6) and how it is the "faith is reckoned (or imputed) for righteousness" it is essential to understand the nature of saving faith. According to Scripture, faith has nothing, no efficacy, in and of itself. Its essence is extraspective as it has been put; it looks away from itself. That is, true faith, faith that saves, does not look at itself and find its confidence in itself but rather it looks away from itself and instead focuses on its object, which it believes and in which it trusts and finds its confidence. Specifically, saving faith trusts in Christ and rests in his righteousness. So, to say, "faith is reckoned for righteousness" is an abbreviated way of saying that Christ's righteousness, the ground of my justification, is reckoned (or imputed) to me for (justifying) righteousness.

A striking instance of this essentially object-bound and object-focused character of faith is Galatians 3:23, 25, where Paul says that with the coming of Christ (see verse 19), "faith has come," even though he has just spoken of Abraham's faith. So tied is faith for its validity to its object, that until the appearance of that object in history (the coming of Christ "in the fullness of time," Galatians 4:4), it is as if faith did not exist prior to that coming. Abraham's faith (the faith of all Old Testament believers) justified because it was focused on the promise that would be fulfilled in the future in the once-for-all work of Christ (his obedience). The faith of New Testament believers justifies as it is focused on the fulfillment of that promise that has already taken place. But apart from that fulfillment (Christ's work) neither is justified and the faith of both is meaningless (see, e.g., 1 Cor. 15:14, 17).

On this issue of faith being imputed for righteousness I would strongly encourage you, if at all possible, especially to read the discussion of John Murray in his commentary on Romans (Eerdmans, 1959), pp. 353-39.

The formulation of the Westminster Confession of Faith in section 1 of its chapter 11 on justification captures well the teaching of Scripture (my emphasis added):

Those whom God effectually calleth, he also freely justifieth: not by infusing righteousness into them, but by pardoning their sins, and by accounting and accepting their persons as righteous; not for anything wrought in them, or done by them, but for Christ's sake alone; nor by imputing faith itself, the act of believing, or any other evangelical obedience to them, as their righteousness; but by imputing the obedience and satisfaction of Christ unto them, they receiving and resting on him and his righteousness, by faith; which faith they have not of themselves, it is the gift of God.