"Friday's Child" is a complicated mess because the Federation wanted mining rights on the planet. If they are going to pump money (or some form of payment for the minerals and services) into that culture's economy, then it seems they see no problem with the PD in this case, and what Kirk is doing is preserving the Federation's position to secure the economic deal and drive off the Klingons.
But the PD is not all-or-nothing. It is where pre-contact societies are concerned, but it also applies to post-contact societies. In that context, it basically means that you can interact with other cultures so long as you don't force your own values and decisions on them, so long as you agree to let them practice their own laws and customs, make their own decisions about internal affairs, etc. (See TNG's "Justice." Why they made contact with such a backward society in the first place is a mystery, but once they had made contact, the Prime Directive still required them to respect local laws, which was why they didn't just break Wesley out and warp away.)
In regards to the TNG Prime Directive maybe, but possibly not in regards to the TOS Prime Directive.
The first reference to the Prime Directive occurs in the episode The Return of the Archons. 
The Prime Directive is explicitly defined in the Star Trek episode "Bread and Circuses," which is set in 2267:
“ "No identification of self or mission. No interference with the social development of said planet. No references to space or the fact that there are other worlds or civilizations." ”
it had been indicated to include purposeful efforts to improve or change in any way the natural course of such a society, even if that change is well-intentioned and kept completely secret. "Pre-warp" is defined as any culture which has not yet attained warp drive technology and is thus, implicitly, unaware of the existence of alien races. Starfleet allows scientific missions to investigate and secretly move amongst pre-warp civilizations as long as no advanced technology is left behind, and there is no interference with events or no revelation of their identity. This can usually be accomplished with hidden observation posts, but Federation personnel may disguise themselves as local sentient life and interact with them.
That is why in Bread and Circuses Kirk comes up with his lie about his ship being somewhere at sea (which I guess JJ is taking literally in Into Darkness
I think the intent in the original series was that Starfleet couldn't go into primitive cultures and mess with them. If a planet managed to figure out there was space travel, and reached out to the Federation, then contact was okay or understood the concepts. If we used our culture now under the original series interpretation, the federation could contact or study Earth. We understand space travel, we are actively searching for life in our own solar system and beyond. If the Vulcan's landed today, it would affect our culture regardless of the technology they had, even if they didn't give it to us, we would eventually figure it out just by dealing with them. In TNG time period I think we would actually have to bump into the Enterprise before they would stop and say Hi. Although in TNG Starfleet was shown studying the Mintakans, which was a very primitive culture, and they got caught. That was a violation, that Picard was trying to fix, which is more along the lines of the original series interpretation.
In Pen Pals the PD became an absolute rule, yet in earlier episodes and later ones as previously mentioned, they argued laws should not be absolute. So I don't know how Picard works that out, given his absolute interpretation of the Prime Directive in Pen Pals, and the mis-interpretation of it in I Borg. When Picard had the chance to wipe out the Borg, he should taken the shot and infected them with the virus that Data and Geordi created. Kind of explains why Enterprise was ferrying diplomats during the Dominion War instead of being on the front lines, because Picard couldn't be counted on to act. He wasn't really trusted after Wolf 359 for a long time.