Sorry to bring bad news, but there simply is no suitable formula, in-place ORing, or list comparison of that type in Numbers.
Of the common workarounds for similar spreadsheet problems, I find the "helper column" approach to be the most straightforward and maintainable. Create a new column to calculate whether or not that row should count in the later summing. The helper column can then be the condition source in your sumif formula, and hidden.
For example, helper-column cell D2 could be:
OR( IFERROR(SEARCH("This*",C2),FALSE) , IFERROR(SEARCH("That*",C2),FALSE) )
and after populating that down the rows, your count formula could be: