Search API module causes server error when saving index fields
After you disable a module (not 'server' ... as in the "update" part of the question ...) and then enable it again, problems like these might get resolved. There are other situations (and modules) where some similar issues occur, and then get resolved via such disable/enable.

To avoid loosing data (like your search that got deleted), this is a possible approach you may want to consider:


look at the module's `.install) file to find out which table it created (during install).
perform a backup of the tables that are mentioned in the .install file (before uninstalling the module.
disable the module, which will also perform the steps you can see within the .install file (for uninstall now) and then enable the module again.
restore your backup of your tables related to the module.