If you enable auto key, then moving mesh vertices will create a key. To set a a key without auto key, click the key button next to the mesh, eg:
Loading Image
It is red here, indicating a key is set on the current frame. It may be yellow (the mesh has been deformed but not yet keyed) or green (the mesh has not been deformed or keyed). Docs:
Key Frames - Spine User Guide: Meshes
Meshes - Spine User Guide: Keying
Using mesh deform keys can be useful, but has drawbacks. First, animating with deform keys just about requires using the straight ahead approach because making changes later is very hard, since the whole mesh is stored for each key. Second, each deform key stores a vertex position for each bone that affects a vertex. Eg, if you have 50 vertices and each vertex is influenced by 2 bones, each deform key stores 50*2=100 vertex positions. For a big mesh with many bones influencing the vertices, this can be a lot of vertex positions for each key. If you animation has many deform keys, this can add up to a lot of data.
An alternative is to weight the mesh to bones, then manipulate the bones to deform the mesh. This is better because only the bones have to be keyed, though you can no longer move each vertex as you can with deform keys. Also, you can show a different mesh (or a linked mesh) weighted to the same bones, then animations that animate those bones will deform that mesh in the same way.