I was watching Dr. Tadashi Tokeida's lecture series on Youtube: https://kevinbinz.com/2017/10/25/isotopy/
For submanifolds $L$ and $K$ being placed in ambient manifold $M$, the overflow can be defined as: $$ \mathbb{o} = (\dim L + \dim K) - \dim M $$
In the case of the Klein bottle can we say because there is only one two-dimensional submanifold inside $\mathbb{R}^3$, the overflow would be: $$ -1 = (2) - 3 \qquad (?) $$
Or since the Klein bottle self-intersecting a curve on the plane in $\mathbb{R}^3$, the overflow should be written as this: $$ 1 = (2 + 2) - 3 \qquad (?) $$
In the former case, there is no self-intersection in $\mathbb{R^4}$ and crossing is possible, because $\mathbb{o}<-1$, but in the latter, there is still a self-intersection at a point in $\mathbb{R}^4$, since $\mathbb{o}=0$.
I would be thankful if someone corrected my mistake.