Site search shows what visitors typed into your own website’s search box: the most common terms, and the terms that grew since the period before. It sits under Visitors in the panel.
Search terms are kept only for the query-string parameters you list in Settings › Measured sites, for example q. Until then both cards say the feature is not configured, which is not the same as “nobody searched”.
Visits that ran a search, by term, with the Parameter it came from, Visits and Share.
Visits, not searches. A visitor who ran the same search six times counts once. The number answers “how many people looked for this”.
Terms ranked by how much more they were searched for than in the period just before, with This period, Before, Change and Movement. The 200 most searched terms across both periods are compared.
From your access log, for sites Loghound reads the log of, and from the beacon, for sites on other servers. They read the same, so a search page on another server appears beside one on this machine. How terms are cleaned and what that means for privacy: Search terms. Both cards export as CSV.