The Web Analytics Associations’ Definitions are useful.
A Count is defined as the most basic unit of measure; a single number, not a ratio.
A Ratio is the relationship between two numbers, one divided by another, and frequently uses the word ‘per’, though, not always.
A Rate is usually the relationship between a variable and time. (Count at t=n / Count at T=n-1) / Count at T=n-1). It’s usually expressed in terms of a percentage.
Be clear about which means what, in which context.
