Does this excerpt from the linear probing Wikipedia page make an assumption?
Yes, it is assumed that probing sequences are identical during the search and insertion. In theory you could use a different probing sequence (e. g. check h(x) index first, then h(x) + 2, and only then h(x) + 1), but it doesn't make any sense - you are better off always using the same sequence to have the minimal expected number of probes.