Do I have this right?...

When it's on an event, it means you can only play one copy of that event per turn.

When it's on any permanent card (unit, enhancement, objective), it means each copy of that card can activate the limited ability once per turn. If such a card should leave and then re-enter play within a turn (such as Echo Base Shield Generator), this counts as two separate copies of the card and therefore allows a second activation.

In either case, the limit starts over at the start of each player's turn, not the full dark-light turn cycle.

It's in the FAQ now.

“Limit once per X” refers to that specific ability on that particular card. Each copy of a card with such a limit may use its ability once per each period of X. This type of limit tends to appear on cards that stay in play.

“Limit 1 per X” refers to all copies of a card, by title. Only 1 copy of a card with the same title may be played by each player during each period of X. This type of limit tends to appear on event cards. If a card or effect with a limit is canceled, it is still considered to have been played or initiated, and counts towards the limit.

And yep, each player does a turn, it's not the full cycle.