"When an upper-class rebel is suddenly disinherited by his fed-up father, he struggles to win his father?s approval in order to have his inheritance restored, but his plans derail when he is forcibly drafted to fight in the second US civil war."

Good points about the two-headed logline. A suggestion that has helped me: backstory is the breeding ground for a good character flaw.

Consider splitting out one the inciting incidents as backstory (probably the disinherited part). Boil it down to a concise character flaw that will make the main action of the story very difficult for the MC in act II, but something he must overcome it in order to achieve the goal.

Spot on what Nir Shelter said about there being essentially 2 inciting incidents: being disinherited and being drafted.

And "the war of 2079" raises other questions that distracts from what might seem to be the premise of the story. The war of 2079? What war? How? Why? It might be better to just avoid mentioning the time frame. Leave that to the story proper. Because the human drama of a son trying to win back his father's approval by joining the military is timeless; it could take place in World War 2, in the Napoleonic Wars, in the Punic Wars for that matter. That it takes place in 2079 is irrelevant for the purpose of a logline unless there is something special, unique about the whatever happens 2079. In which case, it suggests the story is about something else, something greater at stake than a son's personal problem with his father.

Given the constraints of a logline, you really only have space to promote one idea, one plot thread, not two or three. What is the central idea of your story, the principal plot? The struggle to reconcile with the father? Fight in a war? A future civil war that takes place in 2079? What is the core plot around which everything pivots?

You keep making it longer by adding in descriptive phrases and vague notions of unclear actions?I'm gonna go back to my initial suggestion for how to summarize the major events and main character. You can change around certain words or details, but the idea is to tell us only what we absolutely need to know to have a basic understanding of the story's main concepts. If it's more than 25-30 words you're probably saying too much.

Let me put it this way: even a terribly boring or ridiculous idea can be summed up in a simple sentence. "An alcoholic dentist must knit a sweater before sunrise or a psychotic elf will turn his brain to pudding." This would make a really stupid movie, but do you have any questions about what's happening or why? You know the main character and a little something about him, you know his main goal, you know the stakes if he fails, and you know the antagonist. That's all you need. A synopsis would contain more plot detail, such as how this all started, the successes and setbacks he encounters while knitting, how it all turns out?but a logline only needs the very basics, and it must be simple & clear. Don't get fancy with it. An interesting story sells itself; all you have to do tell us what that story is.

Shadow.

There is a recurring problem in each re worded logline you post. You essentially have 2 separate inciting incidents these are:
1 - being disinherited
2 - being drafted into the army

As a result you have 2 major plots crammed into the one logline. You describe both inciting incidents and then describe only one goal for the second inciting incident fighting well in a war.

This makes for the first inciting incident of being disinherited come across as unrelated in motivation to the goal you describe of succeeding in war.

The extended description of the father's reason for disinheriting the MC doesn't add a compelling motivation for the MC. Also the character description of the MC makes him a generally dis-likable person perhaps describe a glaring flaw he can overcome as appose to upbringing circumstances (well off) and unsavory tendencies (trouble making). Perhaps immature or deluded would work better instead.

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REWRITE:
"When a well off trouble-making teenager is disinherited by his father in order to teach him a lesson he struggles and fails to win back his father's approval, but he gains the ultimate chance to prove his worth when he is drafted to fight in the War of 2079."

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( mrliteral, I will not be satisfied until I have your approval. :) )
Thoughts on this one? I'm trying to give the feeling that after the son is disinherited, he kind of sees the error of his ways and he has a eureka moment and tries to gain his father's trust again but fails. Then this incident happens (he is drafted) and he suddenly has a perfect opportunity to show not only his father but himself that he can do something meaningful with his life. Until now he has just been a rich brat with too much time on his hands.
PS. thanks for the comments so far

Agreed with the above.

Also if the MC's inciting incident or in other words the event that motivates him to take action is being disinherited by his father how does that motivate him to achieve something in a war?

The two appear disconnected in terms of cause and effect.

Secondly what is his outer, able to be filmed with a camera and objective goal? This is still unclear from the re draft and as a result it is unclear what the MC will be doing for most of the film.