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The flow past a circular cylinder under diverse conditions is investigated to examine the nature of the different separation mechanisms that can develop. For a fixed cylinder in a uniform, steady, and horizontal stream, the alternating sheddings of vortices, characterizing the Kármán vortex street, occur from two separation points located in the rear cylinder wall. The prediction of the separation positions and profiles is examined in the light of the most recent theory of unsteady separation in two-dimensional flows. It is found that the separation points are fixed in space and located symmetrically about the horizontal axis passing through the center of the cylinder. The unsteady separation profiles are also well-predicted by the theory. If the cylinder rotates on its axis in the anti-clockwise direction, the upper and lower separation points are shifted in the upstream and the downstream direction, respectively, but are no longer attached to the wall and cannot be predicted by the theory. Instead, they are captured as saddle points in the interior of the flow without any connection to on-wall quantities, as suggested by the Moore-Rott-Sears (MRS) principle. The saddle points are detected through a Lagrangian approach as the location of maximum tangential rate of strain on Lagrangian coherent structures identified as the most attracting lines in the vicinity of the cylinder. If, in addition, the uniform stream is unsteady, the Eulerian saddle points, i.e., detected by streamlines, change position in time, but have no direct relation to the true separation points that are defined by Lagrangian saddle points, thus invalidating the MRS principle that is Eulerian by nature. Other separation mechanisms are also described and understood in view of Lagrangian identification tools.