Why MSE formula is looking so different?

Question

Here's what I randomly found on course online (Google Course, Andrew NG, etc.) about Mean Squared Error (MSE)

1. prediction - actual or actual - prediction, It's maybe identical by squared but I never see anyone write F = am instead of F = ma before. Why it's so random swap? (I just curious)
2. The hat -, ~, ^ all are same? (This look like no standard at all?)
3. Some use regular (i),some used sub i, some used super i or (i)

What's actual standard formula I can rely on?

Answer 1

"prediction - actual or actual - prediction, It's maybe identical by squared but I never see anyone write F = am instead of F = ma before. Why it's so random swap? (I just curious)"

The reason noone writes $F=am$ is because $F$ and $a$ are vectors and by convention, you start by the constants ($m$ here), then you add the vector. It's not a random swap, it's just that there is no convention for that, both writing exist.

"The hat -, ~, ^ all are same?" (This look like no standard at all?) Yes, they are all the same, there is no global standard.

"Some use regular (i),some used sub i, some used super i or (i)" The superscript i is applied on specific point $x^i$ while the subscript is applied on the actual value taken by the function $Y$. Some books write $x_i$ or $x_i$ but I have never seen $Y^i$.

"What's actual standard formula I can rely on?" Anyone you like.