How can export the list of Applications I have installed on my Mac as a TextEdit file?

Hold down the option key and select System Information form the Apple menu (the item is normally About this Mac). In the column on the left, select Applications and wait a while. It finds all the applications on your Mac, not just the top level application bundles in /Applications. Choose File/Print and then Save to PDF. You can either use the pdf as-is, or select its entire contents and paste that into a text file.

There doesn't appear to be a way to directly export the System Information utility's output direct to plain text.

ls -l /Applications | open -ef

Get just the app names, and first order sub-folders in /Applications without clutter into TextEdit

$ shopt -s extglob

# get every file/folder in the Applications folder, and then strip path and extension from the app names

$ for x in \( /Applications/* \); do printf "%s\n" "${x//+(*\/|\.*)}";done | open -ef

$ shopt -u extglob