fix(chat): fix custom buttons in playback mode, hide templates in schema chats (Issue #3002, #3013)

[Gimir]

Description:

• improve behaviour of the custom buttons and chat starters in playback mode
• hide message templates in conversations with custom buttons

Issues:

• Issue [Custom buttons] custom buttons are not shown on the first screen and in input message field in Playback chat #3002
• Issue [Custom buttons] Set message template feature should not be available for the conversation with custom buttons #3013

Checklist:

• the pull request name complies with Conventional Commits
• the pull request name starts with fix(<scope>):, feat(<scope>):, feature(<scope>):, chore(<scope>):, hotfix(<scope>): or e2e(<scope>):. If contains breaking changes then the pull request name must start with fix(<scope>)!:, feat(<scope>)!:, feature(<scope>)!:, chore(<scope>)!:, hotfix(<scope>)!: or e2e(<scope>)!: where <scope> is name of affected project: chat, chat-e2e, overlay, shared, sandbox-overlay, etc.
• the pull request name ends with (Issue #<TICKET_ID>) (comma-separated list of issues)
• I confirm that do not share any confidential information like API keys or any other secrets and private URLs

[Gimir]

/deploy-review

Environment URL: https://chat-ai-dial-chat-pr-3035.nightly-test.deltixhub.io
E2E tests status: success