Поведение переменной среды PATH при переключении пользователя с и без входа в систему

Пожалуйста, посмотрите на man bash, опция -l определяет, какой файл конфигурации читается:

INVOCATION
   A login shell is one whose first character of argument zero is a -, or one started with the --login option.

   An interactive shell is one started without non-option arguments and without the -c option whose standard input and error are both connected to  ter‐
   minals (as determined by isatty(3)), or one started with the -i option.  PS1 is set and $- includes i if bash is interactive, allowing a shell script
   or a startup file to test this state.

   The following paragraphs describe how bash executes its startup files.  If any of the files exist but cannot be read, bash reports an error.   Tildes
   are expanded in filenames as described below under Tilde Expansion in the EXPANSION section.

   When  bash is invoked as an interactive login shell, or as a non-interactive shell with the --login option, it first reads and executes commands from
   the file /etc/profile, if that file exists.  After reading that file, it looks for ~/.bash_profile, ~/.bash_login, and ~/.profile, in that order, and
   reads  and executes commands from the first one that exists and is readable.  The --noprofile option may be used when the shell is started to inhibit
   this behavior.

   When a login shell exits, bash reads and executes commands from the file ~/.bash_logout, if it exists.

   When an interactive shell that is not a login shell is started, bash reads and executes commands from /etc/bash.bashrc and ~/.bashrc, if these  files
   exist.  This may be inhibited by using the --norc option.  The --rcfile file option will force bash to read and execute commands from file instead of
   /etc/bash.bashrc and ~/.bashrc.

   When bash is started non-interactively, to run a shell script, for example, it looks for the variable BASH_ENV in the environment, expands its  value
   if it appears there, and uses the expanded value as the name of a file to read and execute.  Bash behaves as if the following command were executed:
          if [ -n "$BASH_ENV" ]; then . "$BASH_ENV"; fi
   but the value of the PATH variable is not used to search for the filename.

   If  bash is invoked with the name sh, it tries to mimic the startup behavior of historical versions of sh as closely as possible, while conforming to
   the POSIX standard as well.  When invoked as an interactive login shell, or a non-interactive shell with the --login option,  it  first  attempts  to
   read  and  execute  commands  from  /etc/profile  and  ~/.profile, in that order.  The --noprofile option may be used to inhibit this behavior.  When
   invoked as an interactive shell with the name sh, bash looks for the variable ENV, expands its value if it is defined, and uses the expanded value as
   the  name of a file to read and execute.  Since a shell invoked as sh does not attempt to read and execute commands from any other startup files, the
   --rcfile option has no effect.  A non-interactive shell invoked with the name sh does not attempt to read any other startup files.  When  invoked  as
   sh, bash enters posix mode after the startup files are read.

   When bash is started in posix mode, as with the --posix command line option, it follows the POSIX standard for startup files.  In this mode, interac‐
   tive shells expand the ENV variable and commands are read and executed from the file whose name is the expanded value.  No other  startup  files  are
   read.

   Bash  attempts to determine when it is being run with its standard input connected to a network connection, as when executed by the remote shell dae‐
   mon, usually rshd, or the secure shell daemon sshd.  If bash determines it is being run  in  this  fashion,  it  reads  and  executes  commands  from
   ~/.bashrc and ~/.bashrc, if these files exist and are readable.  It will not do this if invoked as sh.  The --norc option may be used to inhibit this
   behavior, and the --rcfile option may be used to force another file to be read, but neither rshd nor sshd  generally  invoke  the  shell  with  those
   options or allow them to be specified.

   If the shell is started with the effective user (group) id not equal to the real user (group) id, and the -p option is not supplied, no startup files
   are read, shell functions are not inherited from the environment, the SHELLOPTS, BASHOPTS, CDPATH, and GLOBIGNORE variables, if they  appear  in  the
   environment,  are ignored, and the effective user id is set to the real user id.  If the -p option is supplied at invocation, the startup behavior is
   the same, but the effective user id is not reset.

Два абзаца, которые отвечают на ваш вопрос:

Когда bash вызывается как интерактивная оболочка входа в систему или как неинтерактивная оболочка с параметром --login, сначала она читает и выполняет команды из файла / etc / profile, если этот файл существует. После прочтения этого файла он ищет ~ / .bash_profile, ~ / .bash_login и ~ / .profile в указанном порядке, а также читает и выполняет команды из первой, которая существует и доступна для чтения. Опция --noprofile может использоваться, когда оболочка запускается, чтобы запретить это поведение.

Когда запускается интерактивная оболочка, которая не является оболочкой входа в систему, bash читает и выполняет команды из /etc/bash.bashrc и ~ / .bashrc, если эти файлы существуют. Это может быть запрещено с помощью параметра --norc. Опция --rcfile file заставит bash читать и выполнять команды из файла вместо /etc/bash.bashrc и ~ / .bashrc.