All racists are bigots, but not all bigots are racists

I think that's backward. A racist may have an opinion such as "all black people like rap music", which is based on a racial pivot. That belief is not necessarily predicated with malice or avarice.

That has nothing to do with racism, though. It IS judging based on race, but the implication of racism is that the judgement is harmful to members of that race (rather, the judgement is prejudiced against said members).

In fact, your argument doesn't actually make very much sense. A person isn't racist unless he/she/it has some form of negative belief about members of a given race. The only way this would even matter would be if the racist believed that rap music was ALSO bad, or something along those lines that cast black people in a negative light.

By definition, a racist belief is not necessarily negative. It is only a belief that person holds for an entire race.
A bigot is intolerant whereas a racist might not be. That's why I think the statement is backward.

Merriam-Webster:
Main Entry: rac·ism
Pronunciation: \ˈrā-ˌsi-zəm also -ˌshi-\
Function: noun
Date: 1933
1 : a belief that race is the primary determinant of human traits and capacities and that racial differences produce an inherent superiority of a particular race
2 : racial prejudice or discrimination

Note definition 2 and the 'inherent superiority' clause of definition 1.

That still doesn't require bigotry. If someone were to believe that all Asians are good at math, while a racial stereotype (and therefore racist), it isn't an expression of hatred.
I still say the statement is backward: All bigots are racists but not all racists are bigots.

1 : a belief that race is the primary determinant of human traits and capacities and that racial differences produce an inherent superiority of a particular race

Inherent superiority implies overall superiority implies inferiority of another race. While there may not be hatred, hatred is not necessarily required:

Bigot:
a person obstinately or intolerantly devoted to his or her own opinions and prejudices

Racism distinctly and clearly leads to bigotry, as has been empirically demonstrated in the past.

This is actually two claims. I agree with ryanknapper that the first claim is not true based on the arguments he already made. Maybe it would be more clear that Alkanshel is wrong if we considered a different example. Consider a hypothetical racist who makes the following claim: "I believe caucasians are superior to members of other races, but I may be wrong, and I am willing to consider other views." The statement demonstrates racism under the primary definition given by Alkanshel, but it does not demonstrate devotion to the viewpoint. So, the speaker is a racist but not necessarily a bigot. Since the claim said that all racists are bigots, and we now have a counter-example, it is safe to say that the first claim is false.

Contrary to ryanknapper's position, I believe the second claim is true. A person who is "obstinately or intolerantly devoted to his or her own opinions and prejudices" is a bigot according to Alkanshel's definition even if the opinion or prejudice has nothing to do with race. Take for example the person who is a bigot because of his or her opinion with regard to members of a particular gender, sexual orientation, social class, religion, or religious sect. That person is a bigot, but not necessarily a racist. So, not all bigots are racists.