When studying a hadith, it is best to refer to the details recorded about the same incident in other recorded ahadith. This is so to avoid confusions that may arise from apparent wording of that individual hadith and have a clearer understanding on the matter.
The event of the Thursday also referred to as "hadith of the pen and paper", except that there was no paper in those days, is mentioned in several traditions. In order to understand this event properly we need to read all those traditions as well and then analyze several aspects of the situation. Usually it is only the following tradition or a similar one, that gets quoted individually and discussed and conclusions are drawn:
Narrated Ubaidullah bin Abdullah: Ibn 'Abbas said, "When the ailment of the Holy Prophet became worse, he said, 'Bring for me paper and I will write for you a statement after which you will not go astray.' But Umar said, 'The Holy Prophet is seriously ill, and we have got Allah's Book with us and that is sufficient for us.' But the companions of the Holy Prophet differed about this and there was a hue and cry. On that the Prophet said to them, 'Go away (and leave me alone). It is not right that you should quarrel in front of me." Ibn 'Abbas came out saying, "It was most unfortunate (a great disaster) that Allah's Apostle was prevented from writing that statement for them because of their disagreement and noise." (Sahih Bukhari Hadith 114 - Book of Knowledge - Sunnah.com) - A note from Sunnah.com advises: "It is apparent from this Hadith that Ibn 'Abbas had witnessed the event and came out saying this statement. The truth is not so, for Ibn 'Abbas used to say this statement on narrating the Hadith and he had not witnessed the event personally." See Fath Al-Bari (commentary on Bukhari) Vol. 1, p.220 footnote.
First thing to mention here is that the apparent words of Ibn Abbas (r.a) suggest to a reader that he himself witnessed this event, and Umar (r.a) deliberately and due to some hidden motive did not obey the Prophet ﷺ. However that is not the case when we notice that Ibn Abbas (r.a) is a narrator of the event and not the witness himself, as also noted in Fath Al Bari the best known commentary on Sahih Al Bukhari, by Imam Hajar Al Asqalani. Since Ibn Abbas (r.a) does not experience the situation first hand we should realize that there could be error in our understanding the situation like most of us do when reading how Umar (r.a) reacted.
Second thing to mention here is that the Prophet ﷺ lived for a couple of days after this incident, therefore if he had to issue a decree that he considered being missed, then he would’ve definitely have done it for he knows the promise of Allah ﷻ as given to him before: [Al Quran 5:67] "O Messenger! deliver what has been revealed to you from your Lord; and if you do it not, then you have not delivered His message, and Allah will protect you from the people; surely Allah will not guide the unbelieving people."
Third thing to note here is that even if we accept that Prophet ﷺ was prevented from dictating something then did the argument happening stopped him from doing what he wished to? Obviously no. He indeed did dictate what he wanted to and it was 3 things as mentioned in another tradition below:
Narrated Ibn `Abbas: "Thursday! And how great that Thursday was! The ailment of Allah's Messenger (ﷺ) became worse (on Thursday) and he said, fetch me something so that I may write to you something after which you will never go astray." The people (present there) differed in this matter, and it was not right to differ before a prophet. Some said, "What is wrong with him ? (Do you think ) he is delirious (seriously ill)? Ask him (to understand his state)." So they went to the Prophet (ﷺ) and asked him again. The Prophet (ﷺ) said, "Leave me, for my present state is better than what you call me for." Then he ordered them to do three things. He said, "Turn the pagans out of the 'Arabian Peninsula; respect and give gifts to the foreign delegations as you have seen me dealing with them." (Sa`id bin Jubair, the sub-narrator said that Ibn `Abbas kept quiet as rewards the third order, or he said, "I forgot it.") (Sahih al-Bukhari 4431) 
One very important thing to note is that the 1st tradition mentioned on top ends with the words from Ibn Abbas (r.a) that "It was most unfortunate (a great disaster) that Allah's Messenger was prevented from writing that statement for them because of their disagreement and noise"… which a reader’s mind concludes that Prophet ﷺ was prevented from writing what he wished to and the matter ended. However when see read the 2nd tradition Ibn Abbas (r.a) added words that "Then he ordered them to do three things. He said, "Turn the pagans out of the 'Arabian Peninsula; respect and give gifts to the foreign delegations as you have seen me dealing with them." Therefore jumping to conclusions can be costly.
Same rules should apply when judging the reaction of Hazrat Umar (r.a). It is a known fact that at the time when the Prophet ﷺ requested for a pen and something to write down some advice, those were his last days and due to sickness he was slipping into a state between consciousness and unconsciousness. Hazrat Umar (r.a) and the companions who objected to the Prophet’s request did not do so because they had a hidden agenda or they decided to disobey him as he was passing away. They did it so that there is no discrepancy left in what was going to be handed over looking at the Prophet’s health. Did they do what was wrong? Yes I believe so and every Muslim should say that they were wrong who objected. But did they intend to disrespect the Prophet ﷺ and had a hidden motive? Most definitely not, for those who gave everything away for Islam, for whose truthfulness Quran also bears witness, how could they do such a thing? In fact they proved their loyalty even after the Prophet ﷺ passed away.
Now the question would be what happened to the 3rd thing?
All the traditions narrated by Ibn Abbas (r.a) inform us that he forgot or he did not mention. By observing all the traditions of this event we notice that nowhere do we find what was the third thing. Since destiny did not reveal for us it is not good that we speculate as such would be harmful for us: (Al Quran 49:12) "O you who believe! avoid most of suspicion, for surely suspicion in some cases is a sin."
It has come down to us in the traditions that Laylat ul Qadr night was also hidden for us when the companions argued. The Prophet ﷺ was going to reveal the date of that night but Allah ﷻ prevented him from doing so and this was a blessing as it allows us to worship every night of the last 10 nights and not one night. Prophet Muhammad ﷺ came to educate the people regarding the Night of Al-Qadr (Decree), but two Muslims were arguing with each other. So, he said: “I came out to inform you about the Night of Al-Qadr, but so-and-so were quarreling, so its knowledge was taken away. Yet, perhaps that is for your own good....”[Al-Bukhari] This teaches us that there is divine wisdom in things hidden from our direct knowledge.
Moreover Arabic words that Umar (r.a) used to describe Prophet’s deteriorating health has been translated in English differently in different traditions. A careful observation shows that all those translated words are not disrespecting, they just describe Prophet’s unconscious state. A couple of those traditions are mentioned below:
Umar said: "Verily Allah's Messenger (ﷺ) is deeply afflicted with pain." (Sahih Muslim - Book of Wills) Umar said, "The Prophet (ﷺ) is seriously ill." (Sahih Bukhari - Book of holding fast to the Quran and Sunnah) In another tradition The Companions said, "What is wrong with him? Do you think he is delirious? Ask him (to understand)." (Sahih Bukhari - Book of Jizyah). The word that describes the Prophet’s health in arabic word ahajara أَهَجَرَ, translated as "delirious", comes from the same root as hijrah. Hijrah as all Muslims know means to "migrate from one state to another". So it means that Prophet Muhammad ﷺ was being ushered from this world to the next and that what the companions thought. In fact the worst English translation I've read so far of the term ahajara is "lost his mind’.
This is my input about this event and behavior of Umar (r.a) can only be understood once we understand different aspects of this event.
A Note for the Shia Muslims
At this point it is also important to mention that this hadith is often quoted in Shia gatherings as a "proof" to judge Hazrat Umar's intention and indulge in unnecessary speculation that the third thing must have been designation of Ali ibn Abi Talib as the Prophet's successor. Implying that the Prophet's order was not implemented by his closest and most educated companions. As proven from our discussion that we do not have concrete evidence in regards to the third thing. If this is what some Shia Muslims chose to believe then the onus lies on them to prove this claim. If they cannot, then mere suspicion is a sin (Al Quran 49:12). Also what is very interesting to note is that Shias have been boldly claiming that the Prophet ﷺ designated Ali (a.s) as his successor at the event of Ghadir, when he referred to him as "mawla", more or less a month before Prophet's death. If Ali's successorship was already declared at Ghadir, where thousands were gathered, then why would he need to have it written down for a few persons on his death. Hence this claim contradicts the Shia's famous position regarding event of Ghadir.
I hope this writing helps and may Allah ﷻ guide us all.