There are two issues: One of them is using the same term in different ways. The deeper mistake is believing words have well-defined meanings at all. I think the deeper issue is the more important one. If you solve the deeper issue then the first issue might solve itself.
Yudkowsky’s Disputing Definitions is a good place to start. Other good readings along these lines is How to Do Philosophy by Paul Graham and The Specificity Sequence by Liron.
A good (thought not perfect) definition of confidence is what odds you would bet on. That’s why so many readers of this website are interested in betting markets and the Kelly Criterion.