Wikipedia defines Seigniorage for paper notes and metal coins. Why is seigniorage for paper money not just face value - production cost? I can understand using just the interest rate in the age of the gold standard, but why today? Is it because the Fed is "loaning" the money to the US? So is the Fed not involved in coins?
I think part of the confusion is due to the age of the term and how money has changed over time relative to being backed by precious metals vs using central banks etc.
Historically: Because historically the coin itself was precious metal, if a change occurred between the 'face value' of the coin and value of the precious metal itself, the holder of the coin was less affected since they have the precious metal already in hand. They could always trade it based on the metal value instead of the face value.
OTOH if you buy a note worth an the current price of ounce of gold, and the price of gold goes up, and then the holder wants to redeem the note, they end up with less than an ounce of gold.
In the more modern age The main concern is the cost to borrow funds to put money into circulation, or the gain when it goes out of circulation. The big difference between the two is that bills tend stay in circulation until they wear out, have to be bought back and replaced. Coins on the other hand last longer, but have a tendency to drop out of circulation due to collectors (especially with 'collectible series' coins that the mint seems to love to issue lately). This means that bills issued tend to stay in circulation, while only a percentage of coins stays in circulation. So the net effect on the money supply is different for the two, and since modern 'seigniorage' is all about the cost to put money in circulation, it is different in the case of coins where some percentage will not remain circulating.