RetroGaming

Edit: Complete overhaul of my reply.

In short, this is a Remake of the first Mother 1 game on the NES, but using the modern Earthbound (Mother 2) engine on SNES. So you want to play the original instead. However they provide a second Romhack titled Giygas Strikes Back. I am not sure, but if I understand correctly this is not a Remake, but its EarthBound (Mother 2) with only bug fixes and quality of life improvements applied. If so, maybe this is the ultimate version to play instead the original game. Below is a sum up to the best of my knowledge and understanding.

Original Releases

Famicom / NES               | Mother 1 (1989)       | EarthBound Beginnings (2015, Virtual Console)
Super Famicom / Nintendo    | Mother 2 (1994)       | EarthBound (1995)

Mother 1 is the original first entry in the series on the Famicom. It is released only in Japan, 1989. Much later in 2015 game was first time officially released in English in the US as EarthBound Beginnings on the Wii U Virtual Console.

Mother 2 is the sequel on the SNES 1994. It was also released in the US in English. Because no other game in the series was released prior, they didn't want to name it "2" and changed the title to EarthBound in the US in 1995.

Romhacks

EarthBound Beginnings Remake is a Romhack / Mod of EarthBound on the Super Nintendo. This is a remake of the first game from the Famicom / NES, using the modern engine on the Super Nintendo.

And when I understand it correctly, the second Romhack that is included here named EarthBound Giygas Strikes Back just contains all the bug fixes and improvements, but is not a Remake. So this is basically EarthBound with bug fixes. But I'm not 100% sure.

The only changelog they provide is https://ebbr.neocities.org/EBGSB_changes and titled List of all changes in the quality-of-life patch, EarthBound: Giygas Strikes Back!. But I could not find changelog for the Remake. I'm as much confused as you guys.