This OVA really adds to Emilia's character; it makes S2 even better because you'll see the reason Emilia acts as she does during a certain scene.
Some cut content and clarifications:
The Prologue of the story, which was not adapted, contains some world building about the Four Great Spirits that existed before the story. The Arbiter, Melakuera, who acts to preserve the world’s balance; the Slasher, Zarestia, who indulges in eternal peace in her bed; the Stone, Muspel, a sanctuary that only exists without will nor words; the Sacred Beast, Odglass, who watches over humanity with affection and benevolence and is the sole of the Four friendly to mankind. There is also a mention to Puck as the Beast of the End, an outsider to the Four Great Spirits but with power rivalling them.
The black liquid in Frozen Bonds is NOT the Black Serpent. The black liquid is explicitly stated to be the Black Serpent’s poison, rather than the Black Serpent itself.
This is more of a mistranslation fix rather than cut content, but what Crunchyroll translated as “the four great contaminants” should have been “Four Greats, Outsider”. This is a reference to the unadapted Prologue of the Frozen Bonds LN (see above).
After defeating Melakuera, Puck became the new Great Spirit of Fire. Puck uses fire magic, contrary to popular belief. Fire magic allows you to control temperature, so Puck uses this in order to create ice from moisture in the air.
Without delving into it too much, this is how the magical attributes work:
Fire: Control of temperature.
Water: Mostly healing magic.
Wind: Functions outside of the bodies of living things.
Earth: Functions inside of the bodies of living things
You also have Yin (shadow) and Yang (light).
Yin: Space-time manipulation.
Yang: Buffs.
If you want to know how magic works in Re:Zero, I'd recommend this video by Echidnut: https://www.youtube.com/watch?v=PJPEeFu1uKc. Though be sure to watch if after S2 because it does contain small spoilers for it.
Moreover, Emilia isn't a magic user herself. She uses the spirit-arts, which essentially means that she uses a contracted spirit's power (in this case Puck).
Anyways, S2 is right around the corner, can't wait for your reactions to it. Keep it up!