Reasons and solutions for MySQL sql_mode modification not taking effect

Session 1:
mysql> delimiter //

mysql> create procedure test_for_group_by()
    -> begin
    -> select k,pad,count(*) from test.test group by k;
    -> end //
Query OK, 0 rows affected (0.01 sec)

mysql> delimiter;

Session 1:
mysql> call test_for_group_by();
ERROR 1140 (42000): In aggregated query without GROUP BY, expression #1 of SELECT list contains nonaggregated column 'test.test.k'; this is incompatible with sql_mode=only_full_group_by

Session 2:
mysql> set global sql_mode='STRICT_TRANS_TABLES,NO_ZERO_IN_DATE,NO_ZERO_DATE,ERROR_FOR_DIVISION_BY_ZERO,NO_AUTO_CREATE_USER,NO_ENGINE_SUBSTITUTION';
Query OK, 0 rows affected (0.00 sec)

Session 1:
mysql> call test_for_group_by();
ERROR 1140 (42000): In aggregated query without GROUP BY, expression #1 of SELECT list contains nonaggregated column 'test.test.k'; this is incompatible with sql_mode=only_full_group_by

Session 3:
mysql> show variables like 'sql_mode';
+---------------+------------------------------------------------------------------------------------------------------------------------+
| Variable_name | Value |
+---------------+------------------------------------------------------------------------------------------------------------------------+
| sql_mode | STRICT_TRANS_TABLES,NO_ZERO_IN_DATE,NO_ZERO_DATE,ERROR_FOR_DIVISION_BY_ZERO,NO_AUTO_CREATE_USER,NO_ENGINE_SUBSTITUTION |
+---------------+------------------------------------------------------------------------------------------------------------------------+
1 row in set (0.01 sec)

mysql> call test_for_group_by();
ERROR 1140 (42000): In aggregated query without GROUP BY, expression #1 of SELECT list contains nonaggregated column 'test.test.k'; this is incompatible with sql_mode=only_full_group_by

Session 3:
mysql> select user,host,count(*) From mysql.user group by user;
+---------------+-----------+----------+
| user | host | count(*) |
+---------------+-----------+----------+
| mysql.session | localhost | 1 |
| mysql.sys | localhost | 1 |
| root | localhost | 1 |
| rpl_user | % | 1 |
| test | % | 1 |
+---------------+-----------+----------+
5 rows in set (0.00 sec)

Session 2:
mysql> select routine_catalog,routine_schema,routine_name,routine_type,created,last_altered,sql_mode from routines where routine_name='test_for_group_by';
+-----------------+----------------+-------------------+--------------+---------------------+---------------------+-------------------------------------------------------------------------------------------------------------------------------------------+
| routine_catalog | routine_schema | routine_name | routine_type | created | last_altered | sql_mode |
+-----------------+----------------+-------------------+--------------+---------------------+---------------------+-------------------------------------------------------------------------------------------------------------------------------------------+
| def | test | test_for_group_by | PROCEDURE | 2020-12-24 12:12:10 | 2020-12-24 12:12:10 | ONLY_FULL_GROUP_BY,STRICT_TRANS_TABLES,NO_ZERO_IN_DATE,NO_ZERO_DATE,ERROR_FOR_DIVISION_BY_ZERO,NO_AUTO_CREATE_USER,NO_ENGINE_SUBSTITUTION |
+-----------------+----------------+-------------------+--------------+---------------------+---------------------+-------------------------------------------------------------------------------------------------------------------------------------------+
1 row in set (0.00 sec)

Session 3:
mysql> drop procedure test_for_group_by;
Query OK, 0 rows affected (0.01 sec)

mysql> delimiter //

mysql> create procedure test_for_group_by()
    -> begin
    -> select k,pad,count(*) from test.test group by k;
    -> end //
Query OK, 0 rows affected (0.01 sec)

mysql> delimiter;

mysql> call test_for_group_by();
+--------+-------------------------------------------------------------+----------+
| k | pad | count(*) |
+--------+-------------------------------------------------------------+----------+
| 393975 | 35227182905-15234265621-59793845249-15413569710-23749555118 | 1 |
| 495688 | 09512147864-77936258834-40901700703-13541171421-15205431759 | 1 |
| 497896 | 13152283289-69561545685-52868757241-04245213425-69280254356 | 1 |
| 498573 | 43131080328-59298106536-35954612339-97546855884-75769514803 | 1 |
| 500775 | 27590239742-20204899609-34345212327-79811525340-24267764271 | 1 |
| 501885 | 63188288836-92351140030-06390587585-66802097351-49282961843 | 1 |
| 503330 | 01495266405-82925129145-92643983850-90243995398-18709399387 | 1 |
| 503666 | 40929980986-33813039690-13155419391-97985458477-39771362212 | 1 |
| 504353 | 00505722282-72931248925-57037623248-81117963809-88658076981 | 1 |
| 514246 | 21979564480-87492594656-60524686334-78820761788-57684966682 | 1 |
+--------+-------------------------------------------------------------+----------+
10 rows in set (0.00 sec)

Query OK, 0 rows affected (0.00 sec)