Honestly, the only real issue you'll have is one I've discussed at length with another (former?) user.
"GUIDE DANG IT!"
If your elements are "difficult to intuit", then you need a guide to help figure them out and remember them.
While I had no issue memorizing the vast majority of the Pokemon type chart (at least initially, before we added new types and changed some things around), I'm currently having a VERY HARD time making sense of the type chart in "TemTem".
Pokemon starts you off simple. Fire, Water, Plant. Fire beats Plant, Plant beats Water, Water beats Fire. Easy to understand. Then, it introduces you to two (or three) new elements. Flying, Electric, Bug. It then puts you up against enemies strong/weak to those so you can learn. It slowly introduces the other elements as you move along... finally giving you "Dragon" at the Elite Four. Which, you would likely never know the weaknesses to without the type chart.
TemTem, on the other hand, introduces you to: Crystal, Fighting, Mental. Okay, how does this work? An NPC tells you, but it's difficult to remember. Crystal is strong against Mental (why? no idea). Metal is good against Fighting (okay, reference to Pokemon, makes sense, mind over matter and all). Fighting is strong against Crystal (okay, somewhat makes sense, fighters can smash stone, I guess? Maybe?).
Then, it gets complicated. The area you start in has two new types to teach you. Flying and Neutral. What's strong against Flying? Just Electric. Nothing else. You don't have access to the Eletric Type yet. But, a few trainers you fight do. What's strong against Neutral? Mental. Just mental. You only have access to "Mind" types if you picked the Mind starter. It gets worse. You pick up two more types along the way. Nature and Electric. Electric is the most overpowered Element in the game. It is super effective against 1/3rd of the type chart (4 of 12! Mental, Water, Flying, and Digital!). It is only weak against two elements (Earth and Crystal). Nature is strong against Earth and Water. It's weak to Fire and Toxic (you might have access to Toxic moves/mons later!). Then, you get "Water" types. Water is strong against Earth, Fire and Digital. It's weak to Electric, Nature, and Toxic.
You MIGHT be able to dabble in another element before you beat the first "Zone" as well... "Toxic". What is Toxic good against? Water and Nature. What is its weakness? Just Flying.
You see how complicated that got very quickly? With the correct mons missing, it's difficult to remember any of this off-hand. You need the guide for it.
NO slow ramp up of the types to let you get used to them. No strengths or weaknesses to easily learn. Most of your party will consist primarily of Water/Flying types and the few Electric types they give you early on render much of the rest of the game "a moot point". The chart is difficult to memorize and the design of the game doesn't give you many clues or even the ability to "intuit" anything very easily.
For example: Why is Crystal weak to Fire? I dunno. But, it is! Why is Crystal strong against Mental? NO IDEA! What about Digital? Why is Digital weak to Electric and Water and Itself? Why is it only strong against Mental, Digital, and Fighting?
So, to get back to the topic at hand:
Should Ice and Water be separate or distinct?
I've played it both ways. The biggest issue I run into isn't whether or not they SHOULD be distinct or not... it's how the "type chart" works with it.
My own game separates them out, but there's game lore for that. The power to move and form Ice is different than the power to move and form Water. Ice doesn't have to be water in my game either. It can be any liquid. Even gasses (yep, you can turn a gas into ice and fling it at someone). Water is... just water. It isn't the power to move any and every liquid. Just the power to move one liquid: Water.
There is overlap, but they are not the same. A square is a rectangle, but a rectangle is not a square.
As such, I don't even have the traditional "type chart" either. Each "element" works on a different kind of "beast". Yes, you can use Fire against Nature types, but you're more likely to run into "Dragon" than "Nature" as a typing. You're more likely to run into "Insect" than you would "Fire" as a typing. Put simply, I have two separate charts that don't really interact that much.
Fire, Water, Earth, Air, Ice, Lightning.
Beast, Insect, Dragon, Demon, Plant, Armored, Aquatic, Undead, Angel.
So, with that said, why does one "element" have to be stronger than another? Why have any interaction at all? Is there a need to make Fire weak to Water? Or, can Fire just be strong against Beasts, Insects, and Plants, but weak against Dragons, Demons, and Aquatics?
Separate Water and Ice if you like. Combine them if you like. Just don't make it into a "GUIDE DANG IT!" with your type chart. Keep it simple if you can. Easily intuited if possible.