Standard Controller with Extension in Salesforce vs with sharing extension - which is better?

A controller extension in Salesforce (that extends a standard controller) will look like this in Apex:

public class MyControllerExtension {
  public MyControllerExtension(ApexPages.StandardController ctrl) {..}
  public PageReference actionMethod1() {..}
  public void actionMethod2() {..}
}
and in a VF page as

.. markup

If either the VF page includes an action of the form:
action="{!save}"  // or any of the other standard controller actions 
OR
the controller extension uses apex of the form
this.ctrl.save(); // or any of the other standard controller actions
THEN per the VF doc you cited
those actions will:

executes in user mode, in which permissions, field-level security, and sharing rules of the current user apply. but if the controller extension does its own querying or DML within its action methods or getters, then that logic executes in system context and you are responsible for FLS. Sharing mode on the class definition will determine record visibility in queries