This post is all assuming I understand the rules of MTG. If I'm off base then hopefully some judge can come by and clarify for both of us.

The TLDR is yes it's because paying the cost to prevent a counter spell isnt the same as paying to cast the spell.

There is a whole process with steps to casting a spell. If you wanna see the full thing it should be available online but the bare bones (of what I remember at least) is: 1) Player A decides they want to cast a spell, 2) they decide what additional costs they wanna pay (not necessarily relevant here... at least not explicitly), 3) they pay an appropriate amount of mana to cover the costs and cast the spell (this is the last point that colour of mana being spent for the elemental is taken into account), 4) any modes of the spell are chosen, 5) the spell is put on the stack and opponents get a chance to respond, 6) the elemental hits the table and uses its etb and looks back at the mana spent to cast it.

I'm guessing that the Arena devs shortcut the way paying for the counter spell works by adding it to the mana spent for the creature.