I think that it is necessary to add trace width settings per segment. When routing power traces for example, one net width does not work.

You can change trace width in current version. Specify minimum and maximum trace width in net classes. While routing a trace manually, press W hot key to invoke trace width templates.

Ok, but still, it would be nice to be able to specify trace widths between components as different than the power traces themselves.( in the schematic editor)

This is feature of PCB Layout. Schematic does not include trace structure, just net-list and width can be specified only for the net (net-class).