For example take the elliptic curve corresponding to $y^2 = x^3-30x+56$, http://www.lmfdb.org/EllipticCurve/Q/2304/h/2 says the Kodaira symbol is $III$ at $p=2$ and $I_0^*$ at $p=3$.
I can understand how to calculate it (https://en.wikipedia.org/wiki/Tate%27s_algorithm). (For $p=2$, $2^2$ does not divide $b_8=30^2$ at Step 4 and for $p=3$, Substituting $x+1$ for $x$, I get $y^2 = x^3+3x^2-27x+27$ at Step 2 and $P(T)=T^3+T^2-3T+1 \pmod 3$ has $3$ distinct roots at Step 6.)
What I am interested in but cannot find is its geometric interpretation, such as the pictures in https://en.wikipedia.org/wiki/Elliptic_surface.
I appreciate explanations or good resource for understanding it.
Edit: Thank you for the comment. I recognized that I should learn the geometric in the complex elliptic surface setting, which I am not familiar yet. I will try it.