When I add a new listing in the validity list, it does not keep the new word for later usage.
Is there a “right” way I can add a new value and it will stay in the listing after clicking “save”?.
I cannot reproduce the behavior. Which version of LO do you have?
Also, each cell using a validity list is independent of the other cells using a validity list. Modifying the list for cell A2 keeps the validity list for cell A3 unmodified.
If you are using a cell range for validity, right-click within the existing range and choose Insert then move cells down (or direction of list range). Enter the new entry in the new cell and it will be automatically included
A quick test with a new profile folder reveals that option “Expand references …” is unchecked by default.
Call Tools>Options>Calc>General Options and check option “Expand references when new rows/cols are inserted”.
When you insert new rows (or columns) anywhere below (behind) or within your list, all references will adjust to the new size of your list.
More detailed description of that option: “Expand references when new rows/cols are inserted”
Thanks for all the suggestions.
Villeroy,
Thank you for the best reason as to why and providing the method to fix the problem. 
What is the reason why the variable is originally disabled?
Basically, it was originally unchecked (default). I have no idea as to why it was “…originally disabled”. That’s above my pay grade. LOL
In Calc, goto “Tools - Options - LibreOffice Calc - General”
Under “Input Settings” you will see (if you have not changed it) that
“Expand references when new columns/rows are inserted” is defaulted to “uncheck”. Just “check” it and the Validity listing will works fine for additional listing.
It is one of many wrong decisions about default values. It seems to be rather difficult to get these simple changes through the bug reporting process.
There are reasons to turn this off when it gets in the way while building calculation models. Since virtually everybody tries to build databases on sheets, virtually everybody wants this option to be ON by default.