A notation only involves syntax. A language involves syntax and a shared set of semantics. The language is for humans to communicate with humans in a way that's brief and specific enough that instructions can be reduced to a recipe for the computer.
Higher-level languages don't communicate anything to a computer at all, after all. That's why we need interpretation or compilation.
In the English language, we have a habit of reusing words for similar things or sometimes for almost entirely dissimilar things. A bicycle seat in the US is also known as a "saddle", although there's no animal under it. A car's engine cover in the US is a "hood" and in the UK a "bonnet", although there's no head under it (except that there's a part of the engine called a "head" in a reciprocating piston engine -- in US English, anyway). A group of electrical storage cells grouped together is a "battery" just like a group of cannon which fire together. A CRT or LCD display is called a "monitor", even when used interactively. A "speaker" is used to emit sound even for music or random noise. Given all of these examples (there are many more) it's not surprising that something which people can read, write, and sometimes speak to get a particular, shared idea across is called a "language", even if it doesn't fit a narrow definition otherwise.