https://my.esri.com/#/downloads/Developer%20Tools
I downloaded Python 2.7.8 (32-bit) ArcGIS for Desktop. But, I can't download the updated package into the conda/jupyter environment for use. Am I installing the wrong software?
Each ArcGIS Desktop client (ArcMap and Pro) comes bundled with a Python interpreter. For ArcMap, it is and will only ever be Python 2.x while ArcGIS Pro uses Python 3.x.
Why not use the Python interpreter that is installed with the desktop client?
If you install ArcGIS Pro you will have a conda environment.
In order to use Jupyter notebooks, you need ArcGIS Pro if you intend to use jupyter inside Pro.
Jupyter notebooks and JupyterLab can be used external to Pro but you don't install anything else.
If you are looking for the "arcgis" module, you can install it through the package manager in Pro into a clone (or via conda.
Skip python 2.7, don't install any version of python if you plan to use Pro or arcmap, you are asking for no end of grief