How do I use a "contains" operation as restful contextual filter for text fields
When you expose a filter, you can set a filter identifier.  This can be used to set the filter value via the URL, E.g.:  /search?keywords=foo or  /search?keyworks=foo+bar.

Note that if you need to hide this filter in a user view, you can do that with CSS or the Views Block View Filter module which allows you to move exposed filters to separate blocks that don't need to be placed on a user page.  Not part of your specs.. but nice to know if you need REST and user pages.

Also, given your requirements, you might want to consider either using the Search API module with it's views structure or code your own rest handler. 

The Search API module will lets you easily set up full text searches and manage results order based on boosts, e.g. words found in titles are more important. The DB backend works well in D8, so you don't need Solr server for this.  (Though Solr has a lot of neat features).

Coding your own shouldn't be too hard with the D8 rest implementation.  You can then do any query variations you need.