Your example is a really good one, because it illustrates the power of OR remarkably well. The key is using it to combine several different operators, not just duplicating a single one, e.g. "query site:a.com | site:b.com". That has been my main way of digging deeper over the years, along with "#" to specify a date range.
Interestingly, in case of "query site:firstsite.ee|secondsite.ee" the "|" doesn't seem to work, I get zero results. Why is that? To narrow down the results based on both the domain name and top level domain, I have to add the inurl: operator. E.g. "query inurl:firstsite|secondsite site:.ee"
Strangely, "piping" is the essential thing I do on the Unix command line every day. I haven't put (that) much thought into using Google's "|" operator in a similar vein, that is, to combine several (3-4) different operators. Possibly partly because one used to get pages and pages of results even with very simple, single-operator queries.
Time to refresh my memory about Google's search operators, I guess. Thanks again for sharing this example.