This question is conflating different lighting models, Phong lighting has no concept of dielectrics and the referenced website is clearly just comparing the Phong models to more physically based models. All the websites listed are consistent in their presentation of the Phong model which is fundamentally and intentionally simple as it reflects the capabilities of the hardware it was designed to run on. ( I agree that the flip code site is confusing and the formula's don't even display correctly in my browser)
$I_p$ reads as intensity sub point this is what the entire goal of computing the lighting is. Once intensity is computed for the lighting it is multiplied by the point color and its done. Again, this is for the Phong lighting model. $k_a$, $k_d$ and $k_s$ are constants used to compute $I_p$ that is why they are on the right hand side of the equation. Once $I_p$ is computed $k_a$, $k_d$ and $k_s$ are no longer needed and do not participate in lighting calculations further.
To talk it through...
The term $k_ai_a$ represents the ambient contribution of the light. Think of a room painted bright red with a white light in it. The "ambient" light in the room is going to pick up a red color as it bounces off all those red walls. $k_a$ is most often used to represent that red color while $i_a$ represents the intensity of that light. When multiplied they represent the ambient term for the lighting calculation. $i_a$ is usually just a floating point value and is very small relative to diffuse and specular. Some models will drop the ambient term all together. (but then they are no longer a true Phong lighting model). In real code this term is almost always a pre-multiplied constant and is not actually computed.
$k_d(\hat{L}_m⋅N)i_{m,d}$ here $k_d$ is a constant and again is almost always used as the color of the light, in this example it would be white. $(\hat{L}_m⋅N)$ is the cosine of the angle between the light the surface normal, and $I_{m,d}$ is the intensity of the diffuse light. When multiplied out this term represents the "diffuse" component of the light. In real code the terms $k_d$ and $I_{m,d}$ are almost always a pre-multiplied constant and are not actually computed.
Finally, $k_s(\hat{R}_m⋅\hat{V})^αi_{m,s}$ here again $k_s$ is a constant, and again almost always a color representing the light and $k_s$ almost always is equal to $k_d$. Next $i_{m,s}$ is the specular intensity and it is almost always equal to diffuse intensity, and again is pre-multiplied with $k_s$, is very frequently the same value used for diffuse and is usually constant for a given light. The value $(\hat{R}_m⋅\hat{V})^α$ is the cosine of the angle between view vector and the reflection vector raised to the specular power $\alpha$. Once computed this term represents specular intensity. So in this case the specular $k_s$ is white and the intensity is equal to $i_d$. (so we end up with the same constant used to compute the diffuse term)
Now add them up... A little bit of red light from the ambient plus the diffuse light which is a bright white plus very bright white highlights from the specular. This is $I_p$, call it "light_intensity" now figure out the color of the point, and compute the final color...
// lets make the color a constant blue-green...
vec4 point_color = vec4(0.0,0.2,0.8, 1.0);
final_color = light_intensity * point_color;
