Shia Muslims do not recite Durood-e-Ibrahim in tashahud. But they recite a complete salawaat for the Prophet and his family.
Allahumma salli'ala Muhammad wa al-e-Muhammad.
The Prophet commanded the believers to not only send prayers upon himself but include his family as well. He said: "Do not send an amputated prayer". The companions asked,'What is an amputated prayer?' He said, "When you say 'Allahumma, salli 'ala Muhammad" and stop. Rather, you should say, "Allahumma, salli 'ala Muhammad wa al-e-Muhammad."
> Yanabi' al-Muwaddah, Vo.2, p.59; Ibn Hajar, al-Sawa'iq al-Muhriqah, ch.11, sec.1
Durood-e-Ibrahim is another indication, that the inclusion of the family of the Prophet in the prayer is important.
The link given in the question shows the prayer in beautiful Arabic calligraphy; however, the translation of it is misleading, as al-e-Muhammad and al-e-Ibrahim are translated as followers and not family / progeny which brings a significant difference to the meaning.