the problem is with imprecise wording.
Technically what the elasticity describes is the change of a given variable (say Demand) with an incremental change of a different variable (say Price).
Mathematically this would be the first derivative of the demand function with respect to Price. So if we were precise we would speak about "the elasticity of demand with respect to price"
Because obviously, there may be a ton of other variables in the demand function (say the weather, available income, competitor prices or whatnot). So you could also create many other demand elasticities.
Now, to further complicate matters, prices can be elastic as well - stock prices are almost perfectly elastic, the price of a 100$-bill is perfectly inelastic. Price could actually change as a function of demand (then you've got yourself a set of interdependent equations) or as a function of many other things (like raw material prices or competitor prices). Then you would speak about "the elasticity of price with respect to X".
Unfortunately, business lingo rarely has scientific precision, so usually when people talk about "demand elasticity" and "price elasticity" they mean the same thing. Even if actually, they are two different things.
And yes, you can use demand elasticities (or "price sensitivity", which is another term often used) to determine whether a price increase would be beneficial (the gain in revenue exceeds the loss in customers or quantities bought) or not (customers are very sensitive to price (high demand elasticity with regard to price).
And while we're at it, a few things to keep in mind:
- The elasticity of demand is usually not linear. That's almost always an approximation
- Demand elasticity usually depends on the direction of change - a price raise of 1% (or 1$) does not have the same effect as a price cut of the same magnitude
- Demand elasticity is predictably irrational (read the book by that title, btw.) - a change from 9.99 to 10.00 will have a far greater effect than from 9.98 to 9.99. (Many more examples like this).
Hope this helps,