TLDR: MLK's message was about justice and equality for black people and used non-violence resistance to achieve it.
MLK's message was about justice and equality for black people, and he used certain tactics in order to achieve that goal. His approach was to use non-violent resistance in order to create social change. He believed that if black people were organized enough, they could change our conditions quicker than if we started rioting and tearing shyt up. The reason we think MLK's message was only about integration was because all we were ever taught about him was the I have a dream speech about him wanting to see black and white people able to sit together at the table of brotherhood. X made similar statements after returning from Mecca and establishing the OAAU, but he was still advocating separation from cacs.
Here's a MLK quote about non violent resistance:
That was the tactics he employed in many of his biggest victories. During the Montgomery bus boycott black people stopped riding the bus 100% (non violent resistance). For black people that depended on buses, King and other leaders during the Boycott would raise money, send supplies, and provide alternative transportation:
That was in 1956. By the late 60's King had taken a more "militant" approach to his non violent resistance campaign. It was then he started telling black people to withdraw your money from white owned institutions and put them 100% in black owned institutions.
He also started saying that his next campaign was about repartations for black people
He was assassinated shortly after that video and quote about withdrawing your money from white businesses. He also was working on the poor people's campaign where he was going to bring tens of thousands of people to Washington, DC and create a protest camp in the middle of DC. The camp was supposed to be there permanently and be a 24/7 protest in front of all of the federal buildings about the injustices going on in America for poor people. The poor people's campaign IMO, was going to be where he demanded reparations for black people. It was similar to the occupy wall street movement from a few years back.
As far as Malcolm, here's his words regarding separation vs. segregation:
Personally I think the reason so many people negatively paint MLK as some sellout c00n is because the same people that assassinated him were the same people who's now praising him for his ideas about "peace, love, and brotherhood." Never forget MLK was absolutely HATED in America. White people in the 60's fewed King as Satan incarnate (I've even heard old racist cacs refer to him as Martin Lucifer King). He wasn't loved at all, which is they firebombed his house during the bus boycott and shot him on the balcony in Memphis.
And the same twisting of King's message is what they're currently doing to Malcolm X's message. They're trying to paint X as some pro-integration activist after he came back from Mecca, but he was still very much focused on black nationalism and viewed white supremacy as the enemy.