I went back and found the very first post that contains such a reference. It was by Nick Bostrom and is at:
The language in this post indicates that various processes take place during an OM.
Quoting a small part of the post:
Then, finding that your present observer-moment is at time 0 gives you reason (because of Bayes' theorem) to prefer a hypothesis according to which a larger fraction of all observer-moments are at time 0 to a hypothesis according to which a smaller fraction of all observer-moments are at that time. In the present example, that means that finding yourself at t=0, you should conclude that the chance that both coins will land heads is less than 1/4. This also means that the chance of the first coin landing heads is less than 1/2.
Here we see processes such as discovery, preferring, and concluding taking place within a moment.
This remains common in the language surrounding the idea of OM in the current threads. See for example Stephen Paul King 's composite post raising similar questions at:
Where various authors use processes within an OM such as reference to a memory and thinking.
All this is confusing. How can a process take place within a single moment?
In my view [compressed] is that all possible states of universes preexist [perhaps compressed as interpretable numbers]. The system imbedding these states has a dynamic arising out of the incompleteness of some of its components which randomly provides these states with an instantation of "reality" [being] of indeterminate dimensions. This will give rise to very long strings of states given such being such that the succession of states within the string can be compressed into a few simple rules [such as this string?]. There is no "observation" in this dynamic, but rather just a "flow" [not necessarily steady] of being.