Tarawih Prayer is a confirmed Sunnah that the Prophet (peace and blessings be upon him) performed. Thus, Tarawih was not started by `Umar (may Allah be pleased with him), rather, it was started by the Prophet himself but he did not continue to observe it in congregation so that zealous Muslims would not think that it was obligatory on them.
Elaborating on this, Sheikh Muhammad S. Al-Munajjid states:

Observing Tarawih in congregation in the mosque is better than observing it at home.

This is indicated by the Sunnah and the actions of the Sahabah (may Allah be pleased with them).

1 – al-Bukhaari (1129) and Muslim (761) narrated from `A’ishah (may Allah be pleased with her) that the Messenger of Allah (peace and blessings be upon him) prayed one night in the mosque, and the people followed him in prayer. Then he prayed the next night, and many people came. Then they gathered on the third or fourth night, and the Messenger of Allah (peace and blessings be upon him) did not come out to them. The next morning he said: “I saw what you did, and nothing kept me from coming out to you except the fact that I feared that it would be made obligatory for you.” And that was in Ramadan.

This indicates that observing tarawih in congregation is prescribed according to the Sunnah of the Prophet (peace and blessings be upon him), but he refrained from doing it because he feared that it would be made obligatory for the ummah. When the Prophet (peace and blessings be upon him) died, this reservation was no longer there, because the Sharia was established.

2 – Al-Tirmidhi (806) narrated that Abu Dharr (may Allah be pleased with him) said: The Messenger of Allaah (peace and blessings be upon him) said: “Whoever observes  qiyaam – i.e., tarawih – with the imam until he finishes, it will be recorded as if he spent the whole night in prayer.”

Classified as sahih by al-Albani in Sahih al-Tirmidhi.

3 – Al-Bukhari (2010) narrated that `Abd ar-Rahman ibn `Abd al-Qari said: I went out with `Umar ibn al-Khattab (may Allah be pleased with him) to the mosque one night in Ramadan, and the people were scattered, each man praying by himself. Some men would pray and have groups of people behind them following them. ‘Umar said: “I think if I unite all these people with one reciter, it will be better. Then he resolved to gather them behind Ubayy ibn Ka’b.

Al-Haafiz said:

Ibn al-Teen and others said that ‘Umar based this decision on the Prophet’s approval of those who prayed with him on those nights. Although he disliked that for them, that was based on the fear that it might be made obligatory for them. When the Prophet (peace and blessings be upon him) died, there was no longer any fear of that happening, and ‘Umar thought, because of the potential division that might arise from people praying separately, and because uniting them behind one reciter was more motivating for many people. The majority agreed with ‘Umar’s decision. (Fath al-Baari).

Al-Nawawi said in al-Majmoo`, 3/526:

Praying Tarawih is Sunnah according to scholarly consensus. It is permissible to offer this prayer alone or in congregation, but which is better? There are two well-known opinions on this matter. The correct view according to the consensus of our companions is that observing it in congregation is better. The second view is that it is better to observe it individually.

Our companions said: The difference of opinion has to do with one who has memorized the Qur’an; there is no fear that he may become lazy and neglect it if he prays on his own, and the congregation in the mosque is not going to be affected if he stays away.  But if one of these factors is absent, then praying in congregation is better, and there is no difference of scholarly opinion on this point.

The author of al-Shaamil said: Abu’l-‘Abbas and Abu Ishaq said that observing tarawih in congregation is better than observing it individually, because of the consensus of the Sahabah and the consensus of the scholars of Islam on this point.

Ibn al-Mubaarak, Ahmad and Ishaq favoured praying with an imam during the month of Ramadan.

It is said in Tuhfat al-Ahwadhi in the book of Qiyam (of night prayer):

It was said to Ahmad ibn Hanbal: Do you prefer a man to pray with the people in Ramadan or on his own? He said: He should pray with the people. He said: And I prefer that he should pray with the imam and pray Witr with him. The Prophet (peace and blessings be upon him) said: “If a man prays qiyaam with the imam until he finishes, it will be recorded as if he spent the rest of the night (in prayer).” Ahmad (may Allah have mercy on him) said: “He should pray with the people until he prays Witr with them, and he should not leave until the imam leaves. Abu Dawud said: I saw him (Imam Ahmad) – in the month of Ramadan, praying Witr with his imam, except on one night when I did not attend. Ishaq (may Allah have mercy on him) said: I said to Ahmad: Is praying qiyam in Ramadan in congregation dearer to you or praying on one’s own? He said: I prefer that this prayer is offered in congregation, so as to revive the Sunnah. And Ishaq said the same. (See al-Mughni, 1/457.)

Shaykh Ibn ‘Uthaymeen said in Majaalis Shahr Ramadan, p. 22:

At first the Prophet (peace and blessings be upon him) used to pray Tarawih in congregation in the mosque, then he stopped because he feared that it might be made obligatory upon his ummah…

Then he quoted the two hadiths cited above. Then he said:

“No man should keep away from Tarawih prayer lest he misses out on the reward for it. And he should not leave until the imam finishes Tarawih and Witr, so that he may attain the reward of spending the whole night in prayer.”

Al-Albani said in Qiyam Ramadan: Rather it (praying Tarawih in congregation) is better than praying it alone, because the Prophet (peace and blessings be upon him) did it himself and explained its virtue.  He only did not lead them in praying it in congregation for the rest of the month because he feared that praying at night during Ramadan might be made obligatory for them, and they would be unable to do that, as it says in the hadith of ‘Aa’ishah which is narrated in as-Sahihayn and elsewhere. This concern no longer applied after the Prophet (peace and blessings be upon him) died and Allah had completed Islam. Hence the reason for not praying in congregation when offering night prayers in Ramadan was no longer present, and the previous ruling remained in effect, which is the prescription to offer this prayer in congregation. Hence ‘Umar (may Allah be pleased with him) revived it, as it says in Sahih al-Bukhari and elsewhere.

It says in al-Mawsu`ah al-Fiqhiyyah (27/138):

From the time of ‘Umar (may Allah be pleased with him), the Rightly-Guided Caliphs and the Muslims regularly offered Tarawih prayer in congregation. ‘Umar (may Allah be pleased with him) is the one who united the people in offering the prayer behind a single imam.

Asad ibn ‘Amr ibn Abi Yusuf said: I asked Abu Hanifah about Tarawih and what ‘Umar did. He said: Tarawih is a confirmed Sunnah, and ‘Umar did not base his decision on speculation and he was not introducing bid`ah (an innovation). He did not enjoin it except because of what he knew from the Messenger of Allah (peace and blessings be upon him). ‘Umar introduced this and gathered the people behind Ubayy ibn Ka`b and he offered this prayer in congregation, at the time when the Sahabah – the Muhajirin and Ansar – were still alive, and no one among them objected to that, rather they helped him and agreed with him, and also enjoined it.”