How to change user badge when 'total User Points' increase?
The key part of your question seems to be this:


  I tried using the Rules module, but didn't find a way to compare the current total amount of points of the user to my specifications, (for Rules Event "User was awarded points"), so that based on the result of that comparison, the Rules Action would eventually change the badge assigned to the user.


Indeed, for Drupal 7, there doesn't seem to be some Rules Condition related to the current total user points for a user. Except by using custom PHP code as in this example (which you could use in a Rules Conditionn Execute custom PHP code condition, e.g. to do some type of comparison):

global $user;
$current_points = userpoints_get_current_points($user-&gt;uid, 'all');
if ($current_points &lt; 0) {return TRUE;}


However, there is a Rules Action "Load user points of a user", whereas you can ask for either a specific category, or for all categories together. This Rules Action provides a Rules variable (for which you can specify its name and description) which contains the amount of user points for the requested category. You can then use this variable in subsequent Rules Actions.

If you then want to use that variable within a Rules Condition, you have 2 ways of doing so:


Include the condition in a Rules Component, whereas you trigger the execution of that Rules Component via a Rules Action following the "Load user points of a user" (and passing the variable containing the amount of user points as a variable for the Rules Component).
Use the Conditional Rules module, so that you can specify Rules Conditions within Rules Actions.


Here is a basic example that displays a message about the current total user points for a user, after the user performed an update of some node (= the Rules Event, which you can change to any Rules event that you want, e.g. to "User was awarded points", as in your question):

{ "rules_display_userpoints_after_updating_content" : {
    "LABEL" : "Display userpoints after updating content",
    "PLUGIN" : "reaction rule",
    "OWNER" : "rules",
    "REQUIRES" : [ "userpoints_rules", "rules" ],
    "ON" : { "node_update" : [] },
    "DO" : [
      { "userpoints_rules_get_current_points" : {
          "USING" : { "user" : [ "site:current-user" ], "tid" : "all" },
          "PROVIDE" : { "loaded_points" : { "total_points" : "Number of points in all categories together" } }
        }
      },
      { "drupal_message" : { "message" : "You now have [total-points:value] points" } }
    ]
  }
}


Use the Rules UI to import the above rule in your own site if you want.