First: You've chosen to express "the distance from me to the star" as some formula involving a quantity called $L_0$ --- and then you've asked us to tell you what $L_0$ means. But you're the person who wrote down this expression, so only you can know what you meant by it.
Second: It's very hard to tell what you're actually asking, but I am pretty sure that I can identify the source of your fundamental confusion. Namely: The length of the rod (as defined in your frame, the rod's frame, or any other frame) does not change over time. The distance from you to the star does change over time. You seem to be trying to treat the length of the rod and the distance to the star as perfectly analogous, but they are not analogous at all.
Third: If two observers are both present at an event $E$, and if you know the coordinates that the first observer assigns to an event $F$, you can Lorentz-transform those coordinates to find the coordinates that the second observer assigns to event $F$. You seem to be imagining two observers, one at an event $E$ on the spaceship and one at a different event $E'$ on the star, and trying to Lorentz transform one set of coordinates to the other by blindly applying a formula.
So start over: You are on the ship. Your clock strikes 1:00. Call this event $A$. At that moment (according to you), a clock on the star strikes 1:00. Call this event $X$. At that moment, according to an observer on the star, your clock is striking 2:00. Call this event $B$.
"The distance from you to the star" depends on both the observer and the time when the observation is made. To you, at 1:00 by your clock, "the distance between you and the star" means the distance, in your coordinates, from $A$ to $X$. To you, at 2:00 by your clock, "the distance between you and the star" means something else. To your friend on the star, at 1:00 by his clock, "the distance between you and the star" means the distance, in his coordinates, from $X$ to $B$. Before you can start comparing one of these distances to another, you have to decide which two you're trying to compare.
Finally, with regard to the time it takes light to get from the star to you --- remember that the light currently arriving at your ship left the star at a time when the star was further away than it is now. Light currently leaving the star will arrive at your ship at a time when the star is closer than it is now. So if you want to talk about "the time it takes light to get from the star to your ship", you need to decide not just who's measuring, but which light beam's travel times you're talking about. (Or perhaps you mean some other travel time.)
Bottom line: Different questions have different answers. Questions like "How far does the guy on the star say he is from me?" have many possible meanings, each of which yields a different answer. The key thing you are missing is that you are not being clear (certainly not with us, and I suspect not with yourself) about what you're trying to ask.