There are few warp kinematic models of the Galaxy able to characterise structure and kinematics. These models are necessary to study the lopsidedness of the warp and the twisting of the line-of-nodes of the stellar warp, already seen in gas and dust. We use the Gaia~Data Release 2 astrometric data up to $G=20$mag to characterise the structure of the Galactic warp, the vertical motions and the dependency on the age. We use two populations up to galactocentric distances of $16$kpc, a young (OB-type) and an old (Red Giant Branch, RGB). We use the nGC3 PCM and LonKin methods based on the Gaia observables, together with 2D projections of the positions and proper motions in the Galactic plane. We confirm the age dependency of the Galactic warp, both in positions and kinematics, being the height of the Galactic warp of about $0.2$kpc for the OB sample and of $1.$kpc for the RGB at a galactocentric distance of $14$kpc. Both methods find that the onset radius is $12sim 13$kpc for the OB sample and $10sim 11$kpc for the RGB. From the RGB sample, we find from galactocentric distances larger than $10$kpc the line-of-nodes twists away from the Sun-anticentre line towards galactic azimuths $sim 180-200^{circ}$ increasing with radius, though possibly influenced by extinction. The RGB sample reveals a slightly lopsided stellar warp with $sim 250$pc between the up and down sides. The line of maximum of proper motions in latitude is systematically offset from the line-of-nodes estimated from the spatial data, which our models predict as a kinematic signature of lopsidedness. We also show a prominent wave-like pattern of a bending mode different in the OB and RGB, and substructures that might not be related to the Galactic warp nor to a bending mode. GDR2 triggers the need for complex kinematic models, flexible enough to combine both wave-like patterns and an S-shaped lopsided warp.[abridged]
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