Prepend condabin/conda into PATH. Write the following settings into your shell init files. ~/.bash_profile, ~/.bashrc for Bash, ~/.zshrc for ZSH.
export PATH="/home/user/miniconda3/condabin:$PATH"
Reopen a new shell. Check the result of which -a conda. Make sure the conda from /home/user/miniconda3/condabin takes precedence over other conda executables.
Cleanup and redo conda init.
Open your shell init files. Remove the following content,
# >>> conda initialize >>>
...
# <<< conda initialize <<<
Save the configuration files.
Reopen a new shell, type /home/user/miniconda3/condabin/conda init once.
Done.
Update: But I still recommend you use the Anaconda already installed on your system. There's no need to install two conda packages.
Because conda is a python manger and a package manager. You can create environments with any version of Python you want with command conda.
The only difference between an Anaconda and a Miniconda installation is that, the "base" environment where conda is installed from Anaconda, is bundled with more pre-installed packages than the "base" env from Miniconda.
Configuring conda to store environments under your home directory is enough.
conda config --prepend pkgs_dirs /home/user/.conda/pkgs
conda config --prepend envs_dirs /home/user/.conda/envs
All you need is the execution permission of the command conda, the environments and pkg caches are stored under your home folder.