Scale vs. Precision - Why this definition?
Precision and Scale are based on the mathematical concept of Significant Figures.  You're not alone in that the current definition/usage for these terms is confusing.


  How does the total number of digits relate to the everyday concept of precision (how 'exact' something is measured)?


First, you must remember that everything in computers is represented in binary; however not all numbers are easily represented using binary syntax, specifically when it comes to certain fractions.  One such fraction, ⅓, is actually impossible to represent in a computer.  Even with the largest amount of memory possible, a computer will never be able to accurately represent ⅓ in binary form.  However, it can get more precise (e.g. close) to ⅓ with the more bits (e.g. and therefore decimal places) that are thrown at it.

Precision can therefore be thought of as how exact a number you want to represent.  The more digits the more precise that number becomes.


  How can I think of the number of decimals in terms of scale (how 'big' something is)?


Since Precision is a measure of how exact the number you wish to represent is and we know we're limited in how much we can store, you have to sacrifice precision for scale.  So basically the smaller the scale the larger the final value can become.

Scale may best be explained with how we look at maps.  A map of a state or province shows a much larger scale of area with less detail, whereas a map of a city shows a smaller area at greater detail.

Scale can therefore be thought of as the amount of detail.  The more detail we want to convey comes with the cost of showing less actual stuff.

I don't know if that helps you at all, but hopefully that helps provide some context behind the definitions of these terms.