My association with action RPGs are primarily action games with somewhat minor RPG elements (as in you can't play a racically different character) often with good writing and/or world building.

Deus Ex and other games that fall under the immersive sim umbrella may lean toward action RPG or CRPG, but for me what defines them is deep locational flexibility and choice.

For example, Diablo is clearly an action RPG, but it doesn't really have the location specific elements to it.

Deus Ex is arguably also an action RPG (although less so than most modern action RPGs) while also offering distinct choices and approaches at the location level.

From a branding perspective I get that using just RPG is probably easier and immersive sim is not a well known (or easy to define), category by players.