In this paper we analyse multi-epoch very long baseline interferometry water maser observations carried out with the Very Long Baseline Array towards the high-mass star-forming region AFGL 2591. We detected maser emission associated with the radio continuum sources VLA 2 and VLA 3. In addition, a water maser cluster, VLA 3-N, was detected ~0.5arcsec north of VLA 3. We concentrate the discussion of this paper on the spatio-kinematical distribution of the water masers towards VLA 3-N. The water maser emission towards the region VLA 3-N shows two bow-shock-like structures, northern and southern, separated from each other by ~100mas (~330AU). The spatial distribution and kinematics of the water masers in this cluster have persisted over a time span of 7yr. The northern bow shock has a somewhat irregular morphology, while the southern one has a remarkably smooth morphology. We measured the proper motions of 33 water maser features, which have an average proper motion velocity of ~1.3mas/yr (~20km/s). The morphology and the proper motions of this cluster of water masers show systematic expanding motions that could imply one or two different centres of star formation activity. We made a detailed model for the southern structure, proposing two different kinematic models to explain the three-dimensional spatio-kinematical distribution of the water masers: (1) a static central source driving the two bow-shock structures and (2) two independent driving sources, one of them exciting the northern bow-shock structure, and the other one, a young runaway star moving in the local molecular medium exciting and moulding the remarkably smoother southern bow-shock structure. Future observations will be necessary to discriminate between the two scenarios, in particular by identifying the still unseen driving source(s).