[mysql] When is a timestamp (auto) updated?

If I have a column in a table of type TIMESTAMP and has as default: CURRENT_TIMESTAMP does this column get updated to the current timestamp if I update the value of any other column in the the same row?
It seems that it does not but I am not sure if this is what should happen.
I can not understand what this means (from MySQL documentation):

If the column is auto-updated, it is automatically updated to the current timestamp when the value of any other column in the row is changed from its current value. The column remains unchanged if all other columns are set to their current values. To prevent the column from updating when other columns change, explicitly set it to its current value. To update the column even when other columns do not change, explicitly set it to the value it should have]2

An auto-updated column is automatically updated to the current timestamp when the value of any other column in the row is changed from its current value. An auto-updated column remains unchanged if all other columns are set to their current values.

To explain it let's imagine you have only one row:

-------------------------------
| price | updated_at          |
-------------------------------
|  2    | 2018-02-26 16:16:17 |
-------------------------------

Now, if you run the following update column:

 update my_table
 set price = 2

it will not change the value of updated_at, since price value wasn't actually changed (it was already 2).

But if you have another row with price value other than 2, then the updated_at value of that row (with price <> 3) will be updated to CURRENT_TIMESTAMP.

I think you have to define the timestamp column like this

CREATE TABLE t1 
(
    ts TIMESTAMP DEFAULT CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP
);

See here

Adding where to find UPDATE CURRENT_TIMESTAMP because for new people this is a confusion.

Most people will use phpmyadmin or something like it.

Default value you select CURRENT_TIMESTAMP

Attributes (a different drop down) you select UPDATE CURRENT_TIMESTAMP

Add a trigger in database:

DELIMITER //
CREATE TRIGGER update_user_password 
  BEFORE UPDATE ON users
  FOR EACH ROW
    BEGIN
      IF OLD.password <> NEW.password THEN
        SET NEW.password_changed_on = NOW();
      END IF;
    END //
DELIMITER ;

The password changed time will update only when password column is changed.