This phrasing is in the Ered Luin, Shire, and Breeland deeds. There's something that I'm curious about for doing those runs optimally. There are some quests that are accepted while geographically outside of the Shire's region, but they count as a "Shire" quest in the quest log and for Shire quest deeds. An example of this is the "Mysterious Relic in the Shire" quest, which is accepted in Oatbarton (Evendim), but it counts as a "Shire" quest in the quest log and for the Life of a Bounder deeds. The "Join the Bounders" quest can also be accepted in Ered Luin or Breeland, but it counts as a "Shire" quest in the quest log and deed. There are also some quests that are accepted while geographically inside of the Shire's region, but they aren't counted as a "Shire" quest in the quest log or for Shire quest deeds. An example of this is the "Prologue: Mundo's Complaint" quest, which is accepted in Little Delving, but it doesn't increase the "Life of a Bounder" deed counter.

Which of these count as "accepting a quest or otherwise starting a deed in Shire?" This question applies to the same wording in the Eren Luin% and Bree% categories as well. I'm fine if the answer is either or both. I just want clarification by sutterismine.

As long as the quest contributes to the Locations "master-deed", that will start the timer

For example any quest or deed that contributes to the Deeds of the Shire deed while running Shire% will start the timer. https://lotro-wiki.com/index.php/Deeds_of_the_Shire

It doesn't matter if the quest was completed or gained in the right location.

If it doesn't increase any deeds in the respective region such as the "Prologue: Mundo's Complaint" quest, then you don't have to complete it (unless it's a prerequisite for something that does count of course)