Is Tarawih prayer an innovation that Umar (may Allah be pleased with him) introduced, especially given the Hadith of Aisha (may Allah be pleased with her) which mentions the Prophet's (peace and blessings be upon him) prayer in the mosque without explicit mention of Tarawih or its time?

Tarawih prayer is a confirmed Sunnah. The Prophet (peace be upon him) prayed it congregationally in the mosque for several nights, then he stopped gathering for it out of fear that it might become obligatory for the Ummah. After his passing (peace be upon him), Umar ibn al-Khattab (may Allah be pleased with him) re-established its congregational performance in the mosque behind a single Imam, and this was met with the approval of the Companions.