Separate C2S ecosystem for chat clients, integration and others

I want to share a link to the [list of clients](https://matrix.org/ecosystem/clients/) with somebody I want to chat with if we want to have a chat together and I want to leave the choice of platform and features to them. (See also: the purpose of matrix.to)

Software serving any other purpose must be listed under a different URL.

As the integrations category is already quite loaded, certain types of apps don't fit nicely. For example: moderation bots, admin clients (Matrix Wrench, synadm), game clients (chess) CLI tools to integrate into other software,, voip clients, etc.

As per [discussions](https://matrix.to/#/!CltYqaqkvHpbwjPFcs:matrix.org/$YV0Jw5UGXWgwmaIvopHwNK7pNoXnPzu4LdJHoldPWOM?via=matrix.org&via=element.io&via=envs.net) started on Matrix.

Start by reviewing the current client list at matrix.org/ecosystem/clients/ and the linked Matrix discussion about separating chat clients from other software. Identify how the existing integrations and client listings are organized. Done means chat clients have their own ecosystem URL and other app types have a suitable separate category or location.

Written by the indexing model from the issue text.