I'll address your question on both logical and semantic levels.
Logical discussion
Your confusion about "informative" is due to your inconsistent definition of "informative":
You first say
If determinism is true freewill does not exist
From which you conclude
D's validity actually informs us on freewill
If we let I(x,y) represent the formula "x is informative of y" then your conclusion is equivalent to:
D ⇒ ¬ F ⇒ I(D,F)
Then you say you
Because if D is true I conclude freewill is false. But if D is not true I conclude freewill is false.[...]Naively I'd just conclude the status of D does not inform us about freewill
Now, you are saying:
¬(D ⇒ ¬F ∧ ¬ D ⇒ ¬F) ⇒ I(D,F)
Which is a stronger condition then your first one.
I think this second definition makes more sense for the idea of "informative of" than the simpler "If A then B therefore A informs us of B".
But, there is also a semantic/conceptual issue you may want to address
Semantic discussion
First, let's clarify what we mean by "free will" - even nailing this down has led to a lively (and still ongoing) debate amongst philosophers.
There are different ways to classify the views on free will, but for this question we can think of free will in two broad senses:
Causal viewpoints (How did the agent decide?)
Also called "Libertarian" or "counter-causal" free will, these viewpoints define "free" as in "free from causality". They argue that our actions are "self caused" vs the outcome of physical laws (deterministic or statistical). These views make a distinction between "randomness" and "choice" for an agent. The spontaneous fission of a uranium atom may be "self caused" in the sense that it wasn't determined by prior events, but it's hard to see how an agent randomly choosing things also doesn't rob us of agency.
Note that it is insufficient to characterize this kind of freedom as "Could have chosen otherwise", or something like "Agent S decides to perform action X with the world in state W; however, there are possible worlds where S doesn't perform action X under the same state of W." Both of these conditions are met merely by allowing the agent to act randomly.
Libertarian free will has the challenge of explaining exactly what this "third mode" of action is if it's neither deterministic nor random.
Say I decide to vote for Candidate A because I think they are more likeable. Assume I have counter-causal free will. One could ask then what made this choice free? I presumably acted logically in accordance with my highest-weighted dimension in a candidate (likability). But why did likability rise tot he top as the deciding factor? How did it win out over other dimensions like tax policy, leadership skills, fashion sense?
Well, I freely chose to emphasize likability, and so if Candidate A ends up being a bad leader I am partly responsible for putting them in office due to not choosing to consider a more relevant dimension. It's hard to see how this kind of choice is anything but another instance of determinism or randomness. And so on in infinite regress.
Source viewpoints (Where did the decision originate?)
For these views, an agent is "free" in the sense of being "free from external coercion" (e.g., being held at gunpoint or being controlled by some futuristic technology). So I am free insofar as I can act in alignment with my intrinsic priorities.
Simple example being "I like to have coffee in the morning, so I decide to make a cup of coffee." In this case I am acting with "free will" in this interpretation. It doesn't get into why I like coffee in the morning or if I could have instead chosen to drink tea. I acted in accordance with my desires/goals and hence made a choice freely.
Of course, I could change my preference to tea, either due to some random event in my brain or due to some other chain of causes (e.g., I found a new flavor of tea that is even more enjoyable than coffee).
A challenge for this view is the blurry boundaries around "coerced/forced" since we are not closed systems, but constantly interacting with the world.
Another challenge here is more about moral implications: if we don't choose (in the Libertarian sense) to want what we want, are we really accountable for our actions?
Now on to your argument :)
Is free will independent of determinism?
I see two implicit assumptions in your argument:
1: Free will is non-random and uncaused (Libertarian free will)
2: There can be only two modes of causation in the world: deterministic or random (physicalism/naturalism)
As you can see, the seeds of free will's destruction are present in the above, since (2) rules out the needed ontological space for (1) to be true. So Free will is false regardless of whether we act deterministically or non-deterministically (which here is equivalent to acting randomly).
Under these assumptions, you rightly conclude that determinism is not informative of free will because we've already assumed free will requires non-deterministic, non-random causation self-action.
If we instead take the source-based view of free will, then we will also find that determinism is uninformative of free will because regardless of how we came to our decisions, if we made them without coercion (freely in the source sense) then we are exercising our free will.
If we keep free will as "Libertarian" free will and allow for the possibility of causes that are neither deterministically caused themselves nor random (let's call it an L cause - for Libertarian), then determinism becomes informative:
(1) (L ∨ D) ∧ ¬(L ∧ D)
(2) D ⇒ ¬F
(3) L ⇒ F
(4) if ¬ D:
(4.1) L (1,4)
(4.2) F (3,4.1)
(5) if D:
(5.1) ¬ L (1,5)
(5.2) ¬ F (3,5.1)
Then, with our definition:
¬(D ⇒ ¬F ∧ ¬ D ⇒ ¬F) ⇒ I(D,F)
We see that indeed I(D,F)