On the occasion of the blessed month of Ramadan, the people observe Taraaweeh prayer. My question is: there are some who pray eleven rak’ahs immediately after ‘Isha’ prayer, following the example of the Messenger of Allah (peace and blessings of Allah be upon him); and there are those who pray twenty-one rak’ahs, ten after ‘Isha’ and ten before Fajr, then they pray Witr.

What is the sharia ruling on this way of praying Taraaweeh? Please note that there are those who say that praying qiyaam before Fajr is bid’ah. May Allah reward you with good.
Taraaweeh prayer is Sunnah according to the consensus of the Muslims, as al-Nawawi (May Allah have mercy on him) said in al-Majmoo’.

The Prophet (peace and blessings of Allah be upon him) encouraged Taraaweeh prayer, for example when he said: “Whoever spends the nights of Ramadan in prayer out of faith and in the hope of reward, his previous sins will be forgiven.”

Narrated by al-Bukhaari, 37; Muslim, 760
So how can it be a bid’ah when the Prophet (peace and blessings of Allah be upon him) encouraged us to do it and the Muslims are unanimously agreed that it is mustahabb!
Perhaps the one who said that it is bid’ah meant that gathering to offer Taraaweeh prayer in the mosques is bid’ah.

This too is incorrect, because the Prophet (peace and blessings of Allah be upon him) prayed Taraaweeh in congregation with his companions for many nights, then he stopped doing that lest it be made obligatory upon the Muslims. Then when the Prophet (peace and blessings of Allah be upon him) died and the revelation came to an end, this concern was no longer an issue, because it could not be made obligatory after the death of the Prophet (peace and blessings of Allah be upon him).

So ‘Umar (May Allah be pleased with him) gathered the Muslims to pray Taraaweeh in congregation. The time for Taraaweeh prayer is from after ‘Isha’ prayer until dawn comes. There is no specific number of rak’ahs for Taraaweeh prayer, rather it is permissible to do a little or a lot. The two ways mentioned in the question are both permissible.
That should be based on whatever the people in each mosque think is suitable for them.
The best is that which was proven from the Prophet (peace and blessings of Allah be upon him), which is that he never did more then eleven rak’ahs when praying qiyaam al-layl, in Ramadan or at other times.
Shaykh Ibn ‘Uthaymeen said, after mentioning the number of rak'ahs in Taraaweeh prayer:

The matter is broad in scope. No one should be denounced for praying eleven or twenty-three, because the matter is broader in scope than that, praise be to Allah. Fataawa al-Shaykh Ibn ‘Uthaymeen, 1/407
During Ramadan when we pray salat al-taraweeh, is it allowed to utter the niyyah (intention) aloud?
Uttering the intention out loud when one is going to pray is bid’ah (an innovation), whether that is for Taraaweeh prayers or any other prayer.

Ibn al-Qayyim said in Zaad al-Ma’aad (1/201): When the Prophet (peace and blessings of Allah be upon him) stood up to pray, he would say: “Allahu Akbar,” and he did not say anything before that, or utter the intention (niyyah) out loud at all. He did not say, “I am going to pray such-and-such a prayer, facing the qiblah, four rak'ahs, as an imam or following an imam.” And he did not say “ada’an (on time)” or “qadaa’an (making up a missed prayer)” or “fard al-waqt (the obligatory prayer of this time).”

These are all bid’ahs which were not narrated by any scholar with any isnaad, be it saheeh, da’eef, musnad or mursal … Neither was this narrated from any of the Prophet’s companions, and none of the Taabi’een or the four imams described it as mustahabb.
So the Muslim should have the intention of praying Taraaweeh in his heart only, and not utter anything out loud.
And Allah knows best.