Support new Beaker Integrations API

[satchelbaldwin]

Updates Biome to use the new Beaker Integrations API after the refactor here.

• Moves YAML files over to the new format, which also unfortunately makes the diff huge
• Restructures how examples and attached files are stored to make the editing experience better
• Standardizes the format and uses a new loader that does not use tagging
• Changes context and agent initialization to use the new format that Beaker expects
• Bumps dependencies to align with expected versions
• Fixes docker build to have consistent paths
• Moves adding examples and integrations to exist in Beaker rather than Biome

The pyproject.toml in this release targets Beaker 1.12 which is yet to be released, as this is a breaking change and integrations have been completely reworked.

[mattprintz]

Was deployed via separate PR #61