1863 Civil War Penny?

Hello;
I am a member of a local numismatic association, and last night a member brought in for "show and tell" a part of his collection which he inherited;
It was an 8.5x11 plastic penny page full of "Indian head" pennies;
One was an 1863 "NOT ONE CENT" (with the "NOT" nearly scratched off);
Another was an 1862 "fat" penny;
One member of our club informed him (and me) that the 1862 was indeed legitimate, and shared a little information on these two coins that I have not heard before, or read anywhere.

I have a "blue book" from several years back, and see no indication or notation of coin diameter/thickness changes, and no mention of the "civil war" pennies - could someone please enlighten me on them?

(more info on me - feel free to skip over what follows)

I am first and foremost a penny collector;
As an adult I have come to love collecting silver coinage, and although as a Canadian I don't have access to much American coinage, I quite like the Morgan dollar and Walking Liberty half dollar.
But my first love has always been the penny.

When our government announced in 2011 that the penny was going to be discontinued, I pivoted to start collecting the American penny.

I had a great stroke of luck when a friends aunt permitted me to buy a box of wheat pennies at face value to build out my USA penny collection - I have 1934 to date (with the sole exception of the 1939D penny), and only twenty-eight missing examples for pre 1934 pennies.

I have my USA penny collection in folders, and even have a folder for "Indian Head" pennies - that folder is rather bare... and I don't often have opportunity to purchase them at what I'd perceive as a reasonable cost.

PS: how many of you have all versions of the "regular" 2025 penny?
To date, I've only found a few Denver mint examples.

Thanks in advance for taking the time to reply!!