Core: make Clients and MultiServer compatible to Websockets 14.x

[black-sliver]

What task needs to be completed?

14.x changed the async implementation and has a different API.

In the transition period, we can either use the new implementation on 13.x (not sure how to enable it though), or make non-ported parts explicitly use the legacy implementation that is still available in 14.0.
See 14.0 Changelog and 14.0 Upgrade Guide.

There is no pressure to do it just now since 0.5.1 will stick with Websockets 13.x and there are no known issues with it, but since we'll drop py 3.8 with 0.6.0, it'd be nice to get it done for 0.6.0, hence the Milestone.

• Make MultiServer compatible (either new API or explicit legacy)
• Make MultiServer use the new API/implementation
• Make CommonClient compatible (either new API or explicit legacy)
• Make CommonClient use the new API/implementation
• Check / list other components here

[black-sliver]

zerks volunteered