Within Japan, anime (アニメ) is use as a more common of the word animation (アニメーション). Animation outside Japan is also called this, often called foreign anime or sometimes cartoon.
Outside Japan it is use to only mean animation in Japan (used to be going by as Japanimation) or sometimes animation in the east. There are eastern animations going by there own name such as aeni (South Korea), donghua (China) and more but tend to not be known hence why some people mean anime as eastern animation.
Just like any word that can be changed over time it really depends on the person on what it means to them.