$HOME/bin dir is not on the $PATH

From the top of ~/.profile:

# ~/.profile: executed by the command interpreter for login shells.
# This file is not read by bash(1), if ~/.bash_profile or ~/.bash_login
# exists.
# see /usr/share/doc/bash/examples/startup-files for examples.
# the files are located in the bash-doc package.

So (if you are using bash as your shell) I'm guessing either ~/.bash_profile or ~/.bash_login is on your system. Select one and edit it to include:

export PATH=$PATH:$HOME/bin

Then save and source ~/.bash_login or logout and log back in.

Edit:

You say that both ~/.bash_profile and ~/.bash_login are both missing from your $HOME. I think we need confirm a few things. Please post the results of the following in your original question:

echo $0
echo $HOME
whoami
less /etc/*-release

Edit 2:

Personally, I do not know why ~/.profile is not being included in your case based on the information provided and documentation. While testing I did notice that my ~/.profile is scanned when I ssh in but not when I launch a new terminal.

But, there is a simple solution to allow $HOME/bin to be included in your interactive shell. Edit (create if not present) ~/.bashrc and add the following line to it:

export PATH=$PATH:$HOME/bin

Save, logout and log back in, or source ~/.bashrc.

The export line could be expanded to check that $HOME/bin exists if you like with:

if [ -d "$HOME/bin" ]
then
    export PATH=$PATH:$HOME/bin
fi

Why ~/.bashrc instead of another file? Personally preference and seems to be more reliable too.

The rules for in-sourcing shell startup files are complex. It's likely that with your setup, .profile is not getting included when you open up a new terminal within an X session (try putting an echo .profile inside of .profile and see if the message shows up when you start a shell).

.  "$HOME/.profile"

should reload the profile manually.

Logging in and out of X should also cause .profile to load.

Alternatively, you can do . $HOME/.profile from .bashrc (while using a variable-based guard to prevent double inclusion) to make sure .profile is always included whenever you start a shell.

(You shouldn't need to export PATH as PATH is already an exported variable and modifying its value won't change its export status.)