Random Forest vs. RainForest
Random forest is a learning algorithm. It is an ensemble learning algorithm that uses decision trees as base learners. You wrote the steps for it correctly.

Rain forest is not a learning algorithm. It is an algorithm of constructing a decision tree (how to do splitting) when the dataset is so large that it does not fit the memory. In rain forest, the whole dataset is not required for making a splitting decision. Only some aggregated information (AVC-set for an attribute or AVC-group if you have more memory) is required.

If your dataset is large, and memory is small, you can use rain forest to build several different decision trees. Then use random forest algorithm with those trees as base learners.