I cannot explain why two different translators would come up with different meanings except to say they had different agendas. One agenda, I'm afraid, is the concept that sex is dirty or wrong, and the second is the Christian concept of "original sin." Neither of these is accepted in a Jewish reading of the Hebrew. With JPS translation, it is as follows:
כִּי פְשָׁעַי אֲנִי אֵדָע וְחַטָּאתִי נֶגְדִּי תָמִיד
5 For I know my transgressions, and my sin is always before me.
לְךָ לְבַדְּךָ | חָטָאתִי וְהָרַע בְּעֵינֶיךָ עָשִׂיתִי לְמַעַן תִּצְדַּק בְּדָבְרֶךָ תִּזְכֶּה בְשָׁפְטֶךָ
6 Against You alone have I sinned, and I have done what is evil in Your sight, in order that You be justified in Your conduct, and right in Your judgment. >
הֵן בְּעָווֹן חוֹלָלְתִּי וּבְחֵטְא יֶחֱמַתְנִי אִמִּי
7 Behold, with iniquity I was formed, and with sin my mother conceived me.
Rashi puts verse 7 into context, noting:
Behold, with iniquity I was formed: Now how could I not sin when the main part of my creation was through coitus, the source of many iniquities? Another explanation: The main part of my creation is from a male and a female, both of whom are full of iniquity. There are many midrashim to this verse, but they do not fit the context of the psalm.
conceived me: Heb. יחמתני, an expression of heat, as (Gen. 30:38): “And they came into heat (ויחמנה) when they came to drink.”
Rashi is not saying that sex is bad, but he is saying that, in the wrong context (i.e. outside of marriage and the intent to bring children) it may come from illicit motivations.
Further context is provided by the Mishna at Avot, Ch. 3:1. It quotes Akavia the son of Mahalalel, a rabbi who lived in the first and second centuries, as saying:
"Reflect upon three things and you will not come to the hands of transgression. Know from where you came, where you are going, and before whom you are destined to give a judgement and accounting. From where you came–from a putrid drop [of semen]; where you are going–to a place of dust, maggots and worms; and before whom you are destined to give a judgement and accounting–before the supreme King of Kings, the Holy One, blessed be He."
I should add that a child, from a Jewish point of view, who has not yet experienced signs of puberty, cannot be deemed a sinner because he or she is not yet responsible. In fact, the Talmud in Yavamot states that newborn children start in a pure state. As they grow to learn appropriate behavior, they are capable of sin and good deeds, and their parents bear responsibility for both. People, from our viewpoint, aren't inherrently sinful -- they are imperfect, subject to making bad decisions as well as good ones. As people grow learn Torah, their capacity to make more right decisions in enhanced, and achieving virtual perfection (perhaps as explained by David in Psalm 19) should be our goal always, rather than assuming that "Man Is Lost."