"cheaper price" or "lower price" / "cheaper rate" or "lower rate"

English is not my first language, but I strive to speak it at its best.

Somehow it really grates me when people say that something is at a "cheaper/more expensive price" or "cheaper/more expensive rate". My understanding is that prices and rates can be lower or higher, whilst products/services can be cheaper/more expensive.

I lived by this rule for a while, however today I spot a phrase "cheaper rate" even in an article on BBC website. Look it up if you fancy, search for "Sony is set to sell the PlayStation 4 for a cheaper price than Microsoft's rival Xbox One." Argh!

Please correct me or affirm my understanding. Thank you.

The reason that your are technically correct is this: it is not the price that is cheaper. Price is an attribute of the item being priced. Rather it is the item itself that is cheaper. However, it is perfectly clear that the price attribute can be lower.

Nonetheless. your observation is also correct: "cheaper price", "cheaper rate" etc. are common usage, and it sounds perfectly fine to my native speaker's ear. Strangely, "more expensive rate" is not, in my experience, common at all, and it sounds strange to my ear. Practical English doesn't always follow the rules -- and we can argue over whether that means the rules are incomplete or whether it means that people are too sloppy.

One other thing worth mentioning as a trans Atlantic thing: cheaper in the UK is really primarily about price, however, in the US cheaper tends to have a very strong color of "lower quality" along with the lower price.