python-postgresql, seems to be connected but no re

Question

I am applying SQL commands with Python (PyCharm), but without getting any error, I get no respond from the database. Even if deliberately I use wrong password still no error, and in the run window I get "Process finished with exit code 0". There are two methods that I applied, the first initially worked properly. When I use pgAdmin4, I can apply the SQL commands without any problem. Two block of code are following, and they are for the same method. The first version worked once, but then I encounter what I'm describing above. It mustn't be the absence of commit() command, because I am using 'with' for the connection in the proper way I think. That's for the first block, because in the second I use commit(), with the same outcome.

def save_to_db(self):
    connection = psycopg2.connect(user='postgres', password='valid', database='learning', host='localhost')
    with connection.cursor() as cursor:
        cursor.execute('INSERT INTO users (email, first_name, last_name) VALUES (%s, %s, %s)', (self.email, self.first_name, self.last_name))
    connection.commit()
    connection.close()

And the second one (with commit() instead of using a 'with' for the connection), also without the expected outcome

def save_to_db(self):
    with psycopg2.connect(user='postgres', password='valid', database='learning', host='localhost') as connection:
        with connection.cursor() as cursor:
            cursor.execute('INSERT INTO users (email, first_name, last_name) VALUES (%s, %s, %s)', (self.email, self.first_name, self.last_name))

I would be glad if I could understand what is wrong. Thanks in advance.

edit: Here is the script from which I run the methods:

from user import User    # the class where the methods are
my_user = User('[e-mail]', 'first name', 'last name', None)  # in the database there is auto-increment for the primary key (4th argument)
my_user.save_to_db()