Master rules have gotten a surprising change that has been implemented into the core of the game as of August 2022. Now, refresh damage interrupts all effects taking place, causing the refresh damage point to take priority over any other ability. The official ruling link can be found here.
In terms of practicability, this ruling will make resolving effects faster, letting your opponent perform a random cut if they choose so to let resolve refresh damage earlier rather than letting you complete your chain of effects in resolution before. Undoubtedly, this new refresh may have larger implications in terms of any abilities that are reliant upon either the deck or the waiting room.
Given certain situations, especially when it comes to search abilities, the refresh ruling may negatively impact your deck since a potential target from the deck is lost due to the refresh taking precedence over the resolution. For instance, if a search brainstorm were to land a hit while revealing the last four cards of the deck and/or spilling over into the new deck, the search target you may be wishing for may end up in the clock as the refresh point compared to the old ruling where you would be able to perform a search first and the refresh point last. (Keep in mind that the refresh point will interrupt the reveal effect)
Similarly, due to how the refresh point impacts your clock, refresh point interrupting a salvage ability may provide additional targets if you were to level with the refresh point damage taking place. Any targeting abilities for the waiting room will also resolve after the refresh point, making targets in clock become new targets in your fresh waiting room for the effect as well.
Either way, going forward, keep this new ruling in mind with your games.