Terminal Output Logging Functionality

# Description of the new feature/enhancement
Something that is really missing from the terminal is a logging option for all the output that is currently displayed in the screen. Either plain output or with timestamps of the time the command was issued. It can be saved either in raw format or saving also the colours and other characters from the terminal.

The problem is that there isn't a solid way to log the output on the terminal. On some linux devices you can use the `script` program but it is buggy and requires a lot of changes to make it work as intended. There are other terminal emulators that provide this functionality and it is amazing, as it allows to check all the logs of all sessions. One emulator that is doing this is MobaXterm. Below is an image from that feature settings.


I would like to add an option on the profile selection (or anywhere that is relevant) asking for similar parameters either where the red box is, or on the advanced tab.
The feature should have the option to be enabled by default , select the format (plain or rich text) and the location of the files that will be saved. Also, it would be good to give the option to name the logs with parameters like: timestamp when the session started, username and host (like the title on the terminal windows).
I am not sure if it is duplicate, i searched but i couldn't find a similar issue. If it is please close it

Start by reviewing the profile-selection and advanced-tab settings described in the issue, then identify the existing terminal session and output paths involved. Resolve the open choices around default enablement, plain versus rich output, save location, and filename parameters. Done requires an agreed scope and a working logging feature covering those requirements.

Written by the indexing model from the issue text.