Issues starting database for the first time on Linux

I'm following the instructions for setting up Postgres 11 on my Linux
machine. I'm new to database management, and just need to run a postgres
server to link up to a client on another machine, nothing complicated
(hopefully).

I'm following the long version, but will use the short version of the
documentation to describe where I've got to.

./configure
make
su
make install

> All the above was done through the repository installer on Linux (Manjaro).

adduser postgres

> Not sure why or how, but I found a user 'postgres' already existed on my system, so left it there.

mkdir /usr/local/pgsql/data
chown postgres /usr/local/pgsql/data
su - postgres
/usr/local/pgsql/bin/initdb -D /usr/local/pgsql/data

> No apparent problem here, the text output ends with "Success. You can now start the database server using: pg_ctl -D /usr/local/pgsql/data -l logfile start"

/usr/local/pgsql/bin/postgres -D /usr/local/pgsql/data >logfile 2>&1 &

>I presume this is a similar instruction to the above (just specifying the path to the binary), either way - nothing I try here works

pg_ctl -D /usr/local/pgsql/data -l logfile start - leads to "waiting
for server to start..../bin/sh: logfile: Permission denied stopped
waiting pg_ctl: could not start server"

/usr/local/pgsql/bin/postgres -D /usr/local/pgsql/data >logfile 2>&1 &
- leads to "[1] 2996 [postgres(at)desk ~]$ -bash: logfile: Permission
denied"

postgres -D /usr/local/pgsql/data (starting it in the foreground) - leads to

2019-12-04 10:07:57.604 GMT [3005] LOG: listening on IPv6 address
"::1", port 5432
2019-12-04 10:07:57.604 GMT [3005] LOG: listening on IPv4 address
"127.0.0.1", port 5432
2019-12-04 10:07:57.606 GMT [3005] FATAL: could not create lock file
"/run/postgresql/.s.PGSQL.5432.lock": No such file or directory
2019-12-04 10:07:57.606 GMT [3005] LOG: database system is shut down

Why am I getting permission denied at every turn, I've tried to follow
the instructions to the letter?