Anyway, I'm going to make my own set of definitions, not as a bid to impose them on the rest of the world, nor as an attempt to "interpret" the essential meanings of terms as used in Forge/Story-Games circles. These definitions may be influenced by earlier ones but they're completely de novo. Also, they're just a stepping stone to talking about the underlying concepts as they relate to design and play, in less jargonistic fashion.
System: the formal rules governing the distribution of authority in an RPG, and the transformation of participant declarations into game-world "facts". Systems are concerned with explicit rights and procedures.
Paradigm or ethos: a collection of common or mutually-complementary understandings regarding the responsibilities of the participants and the purpose of play.
Even these definitions aren't so important in their particulars as they allow us to talk about these things separately in design and play. In design, they're fully distinguishable. A rule that says, "The GM may not declare a conflict without the agreement of the players" is formally the same as "the players may veto any conflict proposed by the GM". Both are part of the system. A "rule" that says "The GM should avoid killing player-characters unless they do something stupid" isn't a part of the system, because it doesn't formally address rights or procedures: it doesn't alter the fact that, presumably, the GM has ultimate say over life & death, or at least the right to over-rule the results of other procedures in the system. But it is an attempt to impart or explain a paradigm or ethos.
As a bit of an aside, an explicit paradigm or ethos may or may not be necessary. As I've argued in the past, many games do have a paradigm that guides play even though we tend not to be aware of it. Namely: winning and losing, concepts that are seemingly meaningless outside the "game-space", but which we allow ourselves to care about. The only exceptions to this are activities such as gambling and professional sports, which do have extrinsic outputs that clearly intrude on "real life". But most of the games we play are not of this nature.
Still other games operate on sub-cultural paradigms that barely need explaining to the initiated--and, in any case, can't be fully explained any more than other cultural activities, whose "purposes" and "language" are diverse, and constantly being transformed through use. For example, "going to a club to see a show" has so many possible functions, each understood in varying degrees by different subsets of the club attendees, that one ought to resort to a meta-paradigm of sociality, the idea of a "scene", if one wants to capture the "aboutness" of the activity. (I've never played a LARP, but I'm pretty sure this idea will ring a bell to those who have.) I think it's undeniable that tabletop can have the same quality. In fact most interactions between humans have this quality, but RPGs are one of those activities that can thrive on it. Furthermore there's a wide range between "using an RPG as a general excuse to hang out with friends" and "using an RPG as the focus for a particular mode of socializing". Even if one did seek a method to group the varies "modes" into categories, that would not in itself allow us to directly impart a specific mode.
Finally, some people wish to assert that certain paradigms are "natural" and don't need to be taught. Personally I think this is more likely to be true of "playing pretend" than various varieties of "telling a story", but that's neither here or there: I'm just including this possibility for the sake of completeness. If you can believe that dogs instinctively communicate with barks, growls, and whines--even if they've been separated from "dog culture" since weaning--then maybe it's possible that significant portions of human culture, or its "substrate", are innate and do not need to be taught.
Let's return from the digression. As I said, the distinction between system and paradigm as I define them is absolutely clear when it comes to the designer's job and the rules text itself. A system may be incomplete--for example, it may describe how to resolve combat, without instructing you how to tell if combat occurs: can anyone declare that it's started, or only the GM, or is there some set of objective conditions which automatically triggers combat? But that doesn't stop it from being a system. Implicitly the holes will have to be filled by a paradigm, such as "the group decides collectively based on common sense".
However, once we move to actual play, the system may or may not survive, but the paradigm goes through a complete transformation. It is no longer text, but action, and the difficulties I alluded to with regard to transmission of paradigms now applies much more widely (to virtually all RPGs, if not to all games), regardless of whether the designer made an effort in the "rules text" to impart a paradigm.
The importance of this observation can be seen by briefly returning to the concept of "System" that I've previously dubbed "LP maximalism". Under this concept, it's commonly been noted (usually as an epiphany) that "systemless" or "freeform" RPGs have infinitely complex "Systems" (LP sense) rather than simple ones. But the nature of paradigms in actual play reveals that this is a completely banal assertion: all RPGs work by means of, through, and indeed upon the paradigm, the web of social interactions and understandings, that guide play. A "systemless" game is only "complex" if it requires a drastic shift on the part of the observer: otherwise it's easy as pie.
Conversely, no matter how much or how little system (my sense) a game has, there are very few ways to avoid the complexity of social interaction. One is to sew up as much as possible under formal procedure, or to fall back on very well-worn paradigms like "win/lose". Either way, you impinge on the quality that distinguishes an RPG from a board game. (The effect varies from group to group: if you strongly buy into the notion that "you aren't really playing the game if you never roll the dice", to the point that you're always trying to hammer on the mechanics, then you're more likely to fall into this trap than if you take a light system as an invitation to apply it only when necessary, on top of your largely-freeform style of play. Viz.) Finally, you can pretend the complexity isn't there, either by appealing to naturalism (see "Brain Damage") or by culture-formation and identification.
See also Jim Henley's recent post about the different perspectives on rules, with a dash of polemic from Malcolm Sheppard (eyebeams) in the comments.
Ah, almost forgot: the next step should be to take all this and translate it back into English.