Decouple Compat from new Teams Apps

[rido-min]

This PR removes the dependency from Compat to TeamsApps, so we can decouple the release date of Core/Compat from PA, from the FE release.

• Updates Compat Hosting to Add BotApplication instead of TeamsBotApplication
• Updates CompatTeamsInfo to call each rest endpoint though the ConversationClient.BotHttpClient Instance
• Remove some conversion from Core->Compat by using the proper generic in the REST call

This pull request primarily migrates code from using the legacy TeamsBotApplication and related types in the Microsoft.Teams.Bot.Apps namespace to the newer BotApplication and types from Microsoft.Teams.Bot.Core. It also removes or updates compatibility methods that referenced the old schema, and modernizes dependency registration and usage throughout the codebase. Additionally, there are improvements to the handling of Teams channel account conversions and group conversation logic in the sample bots.

Migration to Bot Core and Cleanup:

• Replaced all references to TeamsBotApplication with BotApplication and updated constructor parameters and dependency injection registrations accordingly in CompatAdapter, CompatBotAdapter, AdapterWithErrorHandler, and related files. ([[1]](https://github.com/microsoft/teams.net/pull/391/files#diff-e6d701ce121dda6651c0dea00498d45e2cc0b0cec446a578f3058be691c7c0f5L26-R25), [[2]](https://github.com/microsoft/teams.net/pull/391/files#diff-e6d701ce121dda6651c0dea00498d45e2cc0b0cec446a578f3058be691c7c0f5L35-R34), [[3]](https://github.com/microsoft/teams.net/pull/391/files#diff-c443812ac75b10f0cb9e0929a1165dee72b7c96adbd232d830a65348ccd1b42bL28-R32), [[4]](https://github.com/microsoft/teams.net/pull/391/files#diff-e5922e6d5c7613f25aefa88e9c33d5a6c34be42d4353354e5f0e36f1715a0290L8-R16), [[5]](https://github.com/microsoft/teams.net/pull/391/files#diff-cb614beeb934fbb598ccac80c19a12a440f301b38b67b0e2796e861754fe59caL274-R280), [[6]](https://github.com/microsoft/teams.net/pull/391/files#diff-4a8e01c7cbf79ebf85f84bb4bef8cf1d3b02dc8a5875df6e9b34ac819fb7f80fL20-R20))
• Updated all using statements and type references from Microsoft.Teams.Bot.Apps and Microsoft.Teams.Bot.Apps.Schema to Microsoft.Teams.Bot.Core and Microsoft.Teams.Bot.Core.Schema as appropriate. ([[1]](https://github.com/microsoft/teams.net/pull/391/files#diff-b9be640e365d9187b0d1a2021dddb84311b737f8c28b16428e3b2a450a4f331aL8-R11), [[2]](https://github.com/microsoft/teams.net/pull/391/files#diff-92ebcca71bdaad244e949c815cbf1152d5a3d814cc007052ae7ad2dd8ab225a9L8), [[3]](https://github.com/microsoft/teams.net/pull/391/files#diff-e6d701ce121dda6651c0dea00498d45e2cc0b0cec446a578f3058be691c7c0f5L9), [[4]](https://github.com/microsoft/teams.net/pull/391/files#diff-c443812ac75b10f0cb9e0929a1165dee72b7c96adbd232d830a65348ccd1b42bL9), [[5]](https://github.com/microsoft/teams.net/pull/391/files#diff-0317e1008e12e306d92289bc421d83933ace0400793ed8ad8abf13b03b68511eL7-R7))

Dependency Injection and Service Registration:

• Changed service registration in InitCompatAdapter.cs and CompatHostingExtensions.cs to use AddBotApplication instead of AddTeamsBotApplication, and updated singleton registrations to use BotApplication. ([[1]](https://github.com/microsoft/teams.net/pull/391/files#diff-cb614beeb934fbb598ccac80c19a12a440f301b38b67b0e2796e861754fe59caL274-R280), [[2]](https://github.com/microsoft/teams.net/pull/391/files#diff-0317e1008e12e306d92289bc421d83933ace0400793ed8ad8abf13b03b68511eL45-R45))

Schema and Compatibility Layer Updates:

• Updated compatibility methods to use ConversationAccount from the new core schema instead of legacy types, including the conversion logic for Teams channel accounts. ([[1]](https://github.com/microsoft/teams.net/pull/391/files#diff-2b447b715e5c577302421ca232985ced9354b64def7b26219525758928d097daL427-L435), [[2]](https://github.com/microsoft/teams.net/pull/391/files#diff-92ebcca71bdaad244e949c815cbf1152d5a3d814cc007052ae7ad2dd8ab225a9L118-L204))
• Removed legacy compatibility methods for meeting info, meeting participants, and channel info that referenced deprecated types. ([core/src/Microsoft.Teams.Bot.Compat/CompatActivity.csL118-L204](https://github.com/microsoft/teams.net/pull/391/files#diff-92ebcca71bdaad244e949c815cbf1152d5a3d814cc007052ae7ad2dd8ab225a9L118-L204))

Bot Sample Improvements:

• Enhanced the EchoBot sample to fetch and echo the user's display name, and in group conversations, to retrieve and display team details and paged member names using the new compatibility APIs. ([core/samples/CompatBot/EchoBot.csL37-R79](https://github.com/microsoft/teams.net/pull/391/files#diff-b9be640e365d9187b0d1a2021dddb84311b737f8c28b16428e3b2a450a4f331aL37-R79))

Miscellaneous:

• Added missing System.Text.Json.Serialization using directive in CompatTeamsInfo.Models.cs. ([core/src/Microsoft.Teams.Bot.Compat/CompatTeamsInfo.Models.csR4](https://github.com/microsoft/teams.net/pull/391/files#diff-b1e303edd8e27c1011150ab48fe66a5de7f2e8ff54a6850a87efc57bcbd46583R4))

These changes modernize the codebase, improve maintainability, and ensure compatibility with the latest Teams Bot SDK core components.

[singhk97]

lgtm