Is it true that the takbir on Eid al-Fitr begins on the eve of Eid and continues until the Eid prayer concludes, whereas on Eid al-Adha it extends throughout the four days?

The Takbir (pronouncement of "Allahu Akbar") on the two Eids is a confirmed Sunnah (definition). On Eid al-Fitr, it begins at sunset on the last day of Ramadan (definition) and ends when the Imam comes out for the prayer, and it is absolute (mutlaq), not restricted by location.

As for Eid al-Adha, the Takbir is both absolute (mutlaq) and restricted (muqayyad). The restricted Takbir occurs after the prayers, from the dawn of the Day of Arafah until the 'Asr prayer on the last day of Tashreeq. The absolute Takbir can be said at all times, from the beginning of Dhul-Hijjah until the end of the days of Tashreeq.