Of course there is an established shared definition. The general consensus of the definitions are:
A remaster takes an existing version of the game and "ports" it, while optimizing it for the newer platforms.
A remake is built from the ground up, often (but not always) using the original as more of a guide and inspiration while often (but not always) adding reinterpretations and new content.
The term "master" comes from the music and film industry and refers to the "final product" of a song or film. It's treated as the "canonical" version of the product, and is optimized toward the platforms it's intended to be experienced with.
When a game is "remastered", the original game is tweaked for a new platform from its original code state.
Remastering a game
Remaking a game
My point is that yes, there is a shared and agreed upon general consensus for the definitions of the words remaster and remake. And if there was some confusion, the
already existing definition of remaster found in the music and film industries give us a good idea.
There are plenty of remakes on the Switch, like Links Awakening, Trials of Mana, Pokemon Lets Go and Advance Wars. Those are all built from the ground up in new engines, and very likely used the originals for direction while adding new content and reinterpreting various aspects.
There are also many remasters of games on the Switch, which include Pokemon BDSP, which evidently use the original game's code according to bugs found in the original DS version cropping up in the Switch version. Other remasters include Metroid Prime Remastered, Super Mario Sunshine (3D all stars), Xenoblade DE, and Skyward Sword HD. All of these games had the original code base updated to be able to run on a newer platform.
So if you want to be sure when you say
@Mekanos is wrong, best thing you could do is prove that SMRPG on Switch is running the original game's code. Otherwise I'm going to go out on a limb and say the game is pretty obviously built from the ground up.