Query regarding functions of postgres

Hi PostgreSQL Community,
Recently I was reading about functions Immutability and security definer
but got confused
Whether the below two functions can be marked immutable or not
1. If a function has constant Raise notice inside it. Eg.

CREATE OR REPLACE FUNCTION text_equals(text, text) RETURNS boolean AS $$BEGIN
RAISE NOTICE 'Comparing two texts';
RETURN $1 = $2;END;
$$ LANGUAGE plpgsql;

2. If a function has Raise notice but extracting current user inside
notice, although its output purely depends on its input arguments eg.

CREATE OR REPLACE FUNCTION text_equals(text, text) RETURNS boolean AS $$BEGIN
RAISE NOTICE 'Current user: %', current_user;
RETURN $1 = $2;END;
$$ LANGUAGE plpgsql;

On security definer part I am confused with the below example

set role postgres;

CREATE OR REPLACE FUNCTION outer_function()
RETURNS TEXT AS $$
DECLARE
user_text TEXT;
BEGIN
SELECT 'OuterFunction() -> Current user is ' || current_user INTO user_text;

user_text := user_text || ' | ' || inner_function();

RETURN user_text;
END;
$$ LANGUAGE plpgsql VOLATILE SECURITY DEFINER;

create role test;

create role alex;

grant create on schema public to test;

set role test;

CREATE OR REPLACE FUNCTION inner_function()
RETURNS TEXT AS $$
DECLARE
current_user_text TEXT;
BEGIN
current_user_text := 'InnerFunction() -> Current user is ' || current_user;
RETURN current_user_text;
END;
$$ LANGUAGE plpgsql VOLATILE SECURITY INVOKER;

set role alex;

select outer_function();

outer_function
-------------------------------------------------------------------------------------------
OuterFunction() -> Current user is postgres | InnerFunction() ->
Current user is postgres

Shouldn't it be "InnerFunction() -> Current user is alex" instead of
postgres as alex called the security invoker function

I tried reading docs but couldn't get any satisfactory answers, it
will be helpful if someone helped me out here

Thanks,

Ayush Vatsa

SDE AWS