the term "independent" is wrong there - but has itself established somehow. - a char that is not in a faction is not independent. - he is part of the NPC faction - just like players who formed up a playerfaction and are bunched together under a player "banner".
the one most important measure to determine a characters affiliation is it ID - and for that, it makes no difference if a player has a player faction tag or not.
so, a player with a LSF ID and a LSF IFF + LSF ship etc. is by no means "independent". he is part of the LSF.
there are a few factions that support independence though - but not cause the player is not part of a playerfaction, but cause the NPC lore suggests independence.
[ zoners, freelancers, mercenaries, generic traders, generic pirates ] - i think thats about it... all other chars are ... well, dependent. playerfactions like to claim authority by impersonating highly ranked characters within the NPC factions hierarchy. - some non-factioned players tried it, too sometimes, but usually fail at that.
anyway, every char that has ... for example ... a LSF ID / IFF is indeed part of the LSF, and it makes no difference if he is part of a playerfaction or not factioned up. only some IDs present some problems ( like the generic military ID ) - another example is ... a char that has a playerfaction tag, but not an appropriate NPC faction tag is not part of the NPC faction. - so, if you see a KNF ship for example that has a military ID means that this char would in fact not be part of the KNF, no matter what the playerfaction tag suggests. ( doesn t happen - and if, its usually only happening for a char-setup ) - but for the matter of the situation, such a char is less a member of the KNF than a non-playerfactioned player with a KNF ID.
but its not a question of being dependent or independent - only the NPC faction itself can make you independent. but yes, i m just having a go at the TERM "independent".