Does elasticity make sense for different levels of prices? If not, how can we fix that? Let's say we know that our customers can't make sense of a raise in 5% of the price. It's too little for them to notice. However, an increase above 10% is easily noticed. If this is true, different level of prices impacts the customers in different ways and the formula we learned at college is not enough for real-world problems. Additionally, 10% of increase in something that costs 10 dollars is different than 10% of increase in something that costs 100 dollars, right?
So, how can we deal with this type of problem? How can we deal with price in real-world? How can we get elasticity in this situation?
The standard elasticity still makes sense. Elasticity is not necessarily constant. Also, I am not sure what sort of formula you learned at your college but elasticity is rigorously defined (and also taught at college level for econ majors) as $EL = \frac{df(x)}{dx}\frac{x}{f(x)}$ for some function $f(x)$ (see Essential Mathematics for Economic Analysis by Sydsæter, Hammond &amp; StrØm - which is an undergraduate textbook used at many econ departments).
For example, for demand function given by: $$Q= a - bp$$ the elasticity will be given by:
$$\epsilon = \frac{dQ}{dp}\frac{p}{Q}= -b \frac{p}{a-bp}$$
so for $a=100$ and $b=10$ it will be:
$$\epsilon = - \frac{p}{100 - 10p}$$
So for example at price 5  dollars the elasticity (in its absolute value) will be 0.1 at price 9 dollars elasticity will be 0.9 and so on. Hence, elasticity can change when prices change and you don't need any new concepts. There is a special type of demand functions where elasticity will always be constant but then there are infinite number of possible demand functions and in most of them it will change with price.