How to restrict SSH users to a predefined set of commands after login?

You can also restrict keys to permissible commands (in the authorized_keys file).

I.e. the user would not log in via ssh and then have a restricted set of commands but rather would only be allowed to execute those commands via ssh (e.g. "ssh somehost bin/showlogfile")

What you are looking for is called Restricted Shell. Bash provides such a mode in which users can only execute commands present in their home directories (and they cannot move to other directories), which might be good enough for you.

I've found this thread to be very illustrative, if a bit dated.

ssh follows the rsh tradition by using the user's shell program from the password file to execute commands.

This means that we can solve this without involving ssh configuration in any way.

If you don't want the user to be able to have shell access, then simply replace that user's shell with a script. If you look in /etc/passwd you will see that there is a field which assigns a shell command interpreter to each user. The script is used as the shell both for their interactive login ssh user@host as well as for commands ssh user@host command arg ....

Here is an example. I created a user foo whose shell is a script. The script prints the message my arguments are: followed by its arguments (each on a separate line and in angle brackets) and terminates. In the log in case, there are no arguments. Here is what happens:

webserver:~# ssh foo@localhost
foo@localhost's password:
Linux webserver [ snip ]
[ snip ]
my arguments are:
Connection to localhost closed.

If the user tries to run a command, it looks like this:

webserver:~# ssh foo@localhost cat /etc/passwd
foo@localhost's password:
my arguments are:
<-c>
<cat /etc/passwd>

Our "shell" receives a -c style invocation, with the entire command as one argument, just the same way that /bin/sh would receive it.

So as you can see, what we can do now is develop the script further so that it recognizes the case when it has been invoked with a -c argument, and then parses the string (say by pattern matching). Those strings which are allowed can be passed to the real shell by recursively invoking /bin/bash -c <string>. The reject case can print an error message and terminate (including the case when -c is missing).

You have to be careful how you write this. I recommend writing only positive matches which allow only very specific things, and disallow everything else.

Note: if you are root, you can still log into this account by overriding the shell in the su command, like this su -s /bin/bash foo. (Substitute shell of choice.) Non-root cannot do this.

Here is an example script: restrict the user into only using ssh for git access to repositories under /git.

#!/bin/sh

if [ $# -ne 2 ] || [ "$1" != "-c" ] ; then
  printf "interactive login not permitted\n"
  exit 1
fi

set -- $2

if [ $# != 2 ] ; then
  printf "wrong number of arguments\n"
  exit 1
fi

case "$1" in
  ( git-upload-pack | git-receive-pack )
    ;; # continue execution
  ( * )
    printf "command not allowed\n"
    exit 1
    ;;
esac

# Canonicalize the path name: we don't want escape out of
# git via ../ path components.

gitpath=$(readlink -f "$2")  # GNU Coreutils specific

case "$gitpath" in
  ( /git/* )
     ;; # continue execution
  ( * )
    printf "access denied outside of /git\n"
    exit 1
    ;;
esac

if ! [ -e "$gitpath" ] ; then
   printf "that git repo doesn't exist\n"
   exit 1
fi

"$1" "$gitpath"

Of course, we are trusting that these Git programs git-upload-pack and git-receive-pack don't have holes or escape hatches that will give users access to the system.

That is inherent in this kind of restriction scheme. The user is authenticated to execute code in a certain security domain, and we are kludging in a restriction to limit that domain to a subdomain. For instance if you allow a user to run the vim command on a specific file to edit it, the user can just get a shell with :!sh[Enter].