Set the iTerm tab title to the current directory, not full path.

current-dir-in-iterm-tab-title.sh

# put this in your .bash_profile
if [ $ITERM_SESSION_ID ]; then
  export PROMPT_COMMAND='echo -ne "\033];${PWD##*/}\007"; ':"$PROMPT_COMMAND";
fi

# Piece-by-Piece Explanation:
# the if condition makes sure we only screw with $PROMPT_COMMAND if we're in an iTerm environment
# iTerm happens to give each session a unique $ITERM_SESSION_ID we can use, $ITERM_PROFILE is an option too
# the $PROMPT_COMMAND environment variable is executed every time a command is run
# see: ss64.com/bash/syntax-prompt.html
# we want to update the iTerm tab title to reflect the current directory (not full path, which is too long)
# echo -ne "\033;foo\007" sets the current tab title to "foo"
# see: stackoverflow.com/questions/8823103/how-does-this-script-for-naming-iterm-tabs-work
# the two flags, -n = no trailing newline & -e = interpret backslashed characters, e.g. \033 is ESC, \007 is BEL
# see: ss64.com/bash/echo.html for echo documentation
# we set the title to ${PWD##*/} which is just the current dir, not full path
# see: stackoverflow.com/questions/1371261/get-current-directory-name-without-full-path-in-bash-script
# then we append the rest of $PROMPT_COMMAND so as not to remove what was already there
# voilà!

Tried that but didn't work. I found a different workaround though. Instead of setting PROMPT_COMMAND, you can add the code that changes the tab name to PS1, like this:
PS1="[\033]0;\W\007]\w $ "
This shows me the full path at the command prompt but only the last directory in the tab name, or ~ if I'm in the home dir. This is perfect for me. Thanks

Thanks @dvbowen!
For clarity, here's what Preferences > Profile should look like:

And .zshrc:

DISABLE_AUTO_TITLE="true"

precmd() {
  # sets the tab title to current dir
  echo -ne "\e]1;${PWD##*/}\a"
}

Success! Thank your

This is just perfect! Thanks

Below is what I am using and it is much simple and memory efficient

DISABLE_AUTO_TITLE="true"
echo -ne "\e]1;\a"

Then open the settings for iTerm2 and select PWD from the title dropdown.

Now the tab title for iTerm2 will look something like below:

Hi, the correct line (for bash) should be:

export PROMPT_COMMAND="${PROMPT_COMMAND:+$PROMPT_COMMAND; }echo -ne \"\033];\${PWD##*/}\007\""

in order to properly append to an existing PROMPT_COMMAND

Source: /etc/bashrc_Apple_Terminal on macOS Ventura

Hi, the correct line (for bash) should be:

export PROMPT_COMMAND="${PROMPT_COMMAND:+$PROMPT_COMMAND; }echo -ne \"\033];\${PWD##*/}\007\""

in order to properly append to an existing PROMPT_COMMAND

Source: /etc/bashrc_Apple_Terminal on macOS Ventura

This fixed the problem I was having, similar to @kenjoegolo above: any time I ran source ~/.bash_profile I would get bash: :echo: command not found, which would persist. Each successive time I ran source ~/.bash_profile, the number of times bash: :echo: command not found displayed would increment (and persist).
Once I replaced the export with @Offirmo 's version, the problem stopped.