Which one of either "where" or "when" works better with "point"?

I'm a non-native English speaker and I have a troubling question. What is the difference in terms of usage of "when" and "where"?

"The fact that my action was so embarrassing singled me out to the point [when/where] my friends asked me if I did so on purpose."

"She was angry to the point [where/when] she punched him and broke his pair of glasses."

There is a similar question online:

The word POINT can refer to a place or a time, therefore a relative adverb WHERE or WHEN can be used respectively leading a relative clause as a modifier. However, in Longman, it says WHERE is used whether the word refers to a place or a time, whereas in Oxford, an example clearly shows: We had reached the point when there was no money left. Please help specify. Thanks.

So does this mean we can use the two interchangably?

The Oxford example is not the same as your example. Their usage of "point" is directly linked to the situation, marking a point in time. Therefore, "when" is appropriate.

In your example, my assumption is that you don't mean "until." Rather, you are talking about the extent of being singled out. When talking about extent, use "where."