What is the ruling concerning vomit (regurgitation) that exits involuntarily from a fasting person, in the event that one causes its exit, and is this considered from involuntary regurgitation?

If the action leads to the expulsion of regurgitation (qalas) into the mouth, with your knowledge or strong presumption that this would occur, then this is considered intentionally inducing vomiting (istisqāʾ), which invalidates the fast according to the majority of scholars. There is no difference between a small or large amount of vomit.

However, if you were compelled to perform the action for fear of harm, then breaking the fast is permissible for you, and you are obligated to make up for it.

If the action did not result in the expulsion of regurgitation into the mouth, then there is no blame upon you. But if it did result in the expulsion of regurgitation into the mouth, your fast is invalidated, and you are obligated to make up for it if it was an obligatory fast. This is not like someone who involuntarily vomits (man dharaʿahu al-qayʾ) but rather is due to an action and inducement from you.

You are not obligated to make up for the past days if you were unaware of the ruling. However, if you were aware of the ruling, then you must make up for all those days.