From a strict risk perspective, particularly if your position sizing is based upon restricting losses to a specific amount, exiting immediately would be the obvious choice from this strict risk perspective.
My following comments below are more focused towards a 'long' term trading system, based on a medium size portifolio ...
Unluckily, exiting from extreme daily volatility hitting a stop (e.g. very bad news ?), and then cutting the trade short 'may' mean missing an otherwise great move if you held on.
This can be particularly difficult in the case, when an entry has quite strict entry parameters, making the initial entry criteria to be met again shortly by the same trade unlikely, so the trade, on a system such as this, means a very premature exit.
Hence why some may wait until the close before making the decision. This could be the eod or eow.
It appears, damn if you do, damn if you don't.
Particularly if you then decide to see a trade to the close, and it continues to head south ... why didn't I exit at the stop ?
Not a moment , am I questioning if signalled , we should not exit a position, the question was do I do it immediately, eod or perhaps eow. With choosing a longer term system, we are perhaps trying to be less sensitive to noise. Was this just noise ? Note, I am not trying to give any answers in this post.
I do not suggest any easy answers, only remark that you have asked an excellent question, and the only suggestion I make is that backtesting may help one be more confident in their decision on what to do, particularly, to then do consistently, once one has made that decision, if trading mechanically.