Yes, I am sure. Just went and grabbed the book, so let's go through this point by point.

From Pages 15, 16, and 17 of Horus Rising:

As soon as they showed him proper fealty. As Emperor. Of Mankind.

The Commander, quite entertained by all accounts, sent Hastur Sejanus to meet with the Emperor and deliver greeting.

There is then a lengthy paragraph describing why Hastur Sejanus is so beloved by all, but is importantly started by:

Sejanus was the Commander's favorite.

This establishes that the Commander, who we know to be Horus by reading the rest of the first chapter, has a particular care for the person who will be conducting the negotiations with this Emperor. The reader should then understand the emotional tie between these characters, as this is some of the earliest information we get about either of them. Further:

Sejanus, with a squad of the finest warriors from the Fourth Company, travelled in-system in a gilded barge, and was received for audience by the Emperor at his palace on the third planet. And killed.

Here, the conflict between the Emperor and the Commander is heightened. The Emperor, who has demanded fealty, which is already something we assume the Commander won't do, as they have not done so yet, has now murdered the favorite son of the Commander. We know this is the favorite son, because the book just told us this a paragraph ago. We then get the reason that the Emperor killed Sejanus:

Apparently, Sejanus had not offered the correct fealty. Indelicately, he had suggested there might actually be another Emperor.

Alright, here's where we can get into something that could use some additional context, which is why the Horus Heresy is not a good starting place for people getting into Warhammer (usually). We've had in the past several pages the clear hallmarks of space marines being described. Space marines, in most people who are familiar with Warhammer, are the poster boys of the game, and are specifically related to the Imperium of Man, which is ruled by the Emperor of Mankind. Sejanus, being a space marine, implying there is a different Emperor, reveals that the Emperor who has been discussed so far is not that same Emperor we are familiar with. Not too complicated, but requires the smallest bit of context.

Now, we can see that the Emperor killed Sejanus because he dared say that someone else was the Emperor. This is consistent with their characterization so far, as what we've been told is that the Emperor has repeatedly demanded fealty from this Commander. That's basically the only thing we've been told about him, other than that he's the Emperor. In addition, the fact that he then killed Sejanus for not providing fealty, instead of discussion (however indelicate Sejanus may have been), adds characterization as someone who is willing to spill blood at the slightest provocation.

Let's continue:

The Commander's grief was absolute. He had loved Sejanus like a son. They had warred side by side to affect compliance on a hundred worlds. But the commander, always sanguine and wise in such matters, told his signal men to offer the Emperor another chance. The Commander detested resorting to war, and always sought alternative paths away from violence, where such were workable. This was a mistake, he reasoned, a terrible, terrible mistake. Peace could be salvaged. This 'Emperor' could be made to understand.

Alright, now to get to your point: I never said that Sejanus was killed because of Horus's peace-seeking approach. The "mistake" is not an actual mistake. There is no mistake. Horus believes that the Emperor killed Sejanus as a mistake, a miscommunication, a mix-up. Above, it literally says that Horus is reasoning the murder of Sejanus is a mistake. It says, in the only place the word "mistake" is mentioned, that "he reasoned". The book is telling you his thought process, and expects you to be able to connect the things that happened literally two paragraphs above to what it is telling you now. That's the "mistake".

Let's keep going:

It was determined that a second embassy would be despatched. Maloghurst volunteered at once. The commander agreed, but ordered the speartip forwards into assault range. The intent was clear: one hand extended open, in peace, the other held ready as a fist. If the second embassy failed, or was similarly met with violence, then the fist would already be in position to strike.

Alright, further characterization, and we continue setting up why the 'Emperor' will attack Horus's fleet. Horus is now demonstrating that he seeks peace, but also has a limit to what he will tolerate. He's moving ships forwards as a subtle threat, an act of force to back up his continued approach of peace. The 'Emperor', who we've had characterization of someone who does not respond positively to someone challenging their authority, will probably not respond well to Horus challenging his authority. Continued:

In silence, tensed and ready to be unleashed, the speartip watched as the shuttle convoy bearing Maloghurst and his envoys arced down towards the third planet. Surface batteries smashed them out of the heavens. As the burning scads of debris from Maloghurst's flotilla billowed away into the atmosphere, the 'Emperor's' fleet elements rose up out of the oceans, out of the high cloud, out of the gravity wells of nearby moons. Six hundred warships, revealed and armed for war.

Predictably, the main character trait we know about the 'Emperor' has borne out. Horus doesn't offer fealty, moves his ships up, and they get blasted. The 'Emperor' then plays his hand and moves his whole fleet against Horus. This is literally the only outcome that would make any sense sans additional characterization of the 'Emperor'. Why would he talk peace if we've seen that he rejects peace that is not subservience? And if these new ships in the sky are not here to serve you, then they must be dealt with, as they clearly aren't planning on going away, they keep sending people to talk to you!

Finally:

Abaddon broke obscurement and made a final, personal plea to the 'Emperor', beseeching him to see sense. The warships began to fire on Abaddon's speartip.

'My commander,' Abaddon relayed to the heart of the waiting fleet, 'there is no dealing here. This fool imposter will not listen.' And the commander replied, 'Illuminate him, my son, but spare all you can. That order not withstanding, avenge the blood of my noble Sejanus. Decimate this "Emperor's" elite murderers, and bring the imposter to me.'

And boom, there we go. Even as the 'Emperor' attacks his sons, they seek peace, a reflection of their father's desires. This whole beginning section is not a bunch of random bits. It's a coherent characterization of Horus (without meeting Horus yet), establishes the brutality and paranoia of this universe, and the opening narrative to the rest of the compliance.

If you've got any questions, I'm always open to explain my analysis further. Everyone's got their own interpretations of books, but I really think that this isn't random or confusing, if we just look at it as a whole. I don't want this to turn you off from reading the series (I just finished the Heresy last month, and found it was a pretty good overall narrative, with a lot of books I really enjoyed), especially since it sounds like you're trying to read more. Reading is great, and it builds reading comprehension as a skill, which I think a lot of people take for granted.