Type 1 is where is the colonizers set up new outposts in more-or-less unoccupied land. This happened in Hong Kong and Calcutta. This is usually benign for the host country.
Type 2 is where the colonizers steal the land from the natives and occupy it themselves. This happened in the United States and to some extent in Ireland. This type of colonialism is usually terrible for the natives.
Type 3 is where the colonizers install themselves as a ruling caste, but otherwise let the natives be. They extract taxes and provide governance in return. This was the case in India. It was also the case when the Normans invaded England, the Spanish invaded Mexico, or the British took over India.
Type 3 often works out better for the natives, because the existing regime had to be pretty backward and corrupt in order to fall to a small band of foreigners. The foreigners install themselves as competent stationary bandits and the country at least enjoys some stability and rule of law.
And how do you even move a culture forward? It sounds like you're implying that some cultures are better than others. Please tell me you're not saying that.
So you're saying Aztec culture wasn't worse than 19th century British culture? Cultures that had human sacrifice are equal to cultures that perfected the steam engine and flush toilets? Please tell me you are not that willfully ignorant.
It should be blindingly obvious that certain elements of culture are worse than others, unless you are willing to argue that religions with human sacrifice are just as good as religions without human sacrifice, or that funerals with Sati are just as good as funerals without Sati. If certain elements of culture are better or worse, than it would be a miracle if all cultures exactly balanced out so that the positive elements and negative elements were equal, thus making all cultures equal to each other.