For moving a feature, the process is to edit the layer, use the "move feature" button, then drag and drop the feature.
I have a point shapefile layer, and would like to move a set of points. This video does exactly what I need to do at this timestamp: https://youtu.be/yiihopTnewg?t=50
However, in my case, there is no "Move feature" button. I can use the "Edit vertex" button, but I can only move one point at a time, and have to select in a different manner to the normal yellow selection process.
In the image below, I want to move the selected features to align with the square.