Hi @ravenna
The first link you posted doesn't work.
Try searching for [https://www.google.com/search?client=firefox-b-d&q=ec003+battery|ec003 battery] (examples only) to find suppliers that suit you best.
This is the model number of the battery used in the iPod mini.
Here's the ifixit [guide|411] guide that may help.
As to recommendations for suppliers, ifixit discourages this (except if they're for very hard to find parts) as it leads to an increase in spam being posted.
When you find a supplier (or two or three) that suit, check the reviews online regarding their service, returns policy, guarantee etc and then use your instincts as to which one you prefer to deal with.