In the above mentioned quote from docs, client means the client programmer, who is using the oracle code in this case. and could be using an API developed by you or a third party. It does not mean the user of the program as you are assuming.
If you are making an API that can/would be used by client programmers, who may not have direct contact with you (common in opensource/ github projects), the client programmer would look only at the java docs to see what type of exceptions are being generated.
With checked exceptions the client have to handle exception
With Unchecked exception the client has an option to handle or leave handling of exception. NumberFormatException would be thrown by Java Runtime. It would propagate to the runtime level. The client programmer may choose to handle this unexpected situation or he may leave it to the Runtime environment.
Note that exceptions are not a means to control program flow. but a means to handle unexpected situations (alarm client programmer in this case) .