It sounds like you want Zapier to find not only which record to update, but also identify which field in the record to update. The search functionality that you linked to will only help find which record to update. It will not identify which field (column) to update.
In order to update different fields, you need to use different paths in Zapier, one path per field that could be updated. Setup a formula field in Airtable to identify which field is the next available field. Then have the Zap take a different path based on the formula field value.
Note that paths in Zapier require a professional plan or higher, and you are limited in the number of paths you can create.
You can also look into using Integromat instead.
Or if you want to use Zapier but do not want to pay for a professional plan, you could combine Zapier with an Airtable automation script. Have Zapier put the value in a “temporary/staging” field. Then have an Airtable automation script copy the value into the proper column.