Но как мне отловить данные "start"?
Deep linking
Telegram bots have a deep linking mechanism, that allows for passing additional parameters to the bot on startup. It could be a command that launches the bot — or an auth token to connect the user's Telegram account to their account on some external service.
Each bot has a link that opens a conversation with it in Telegram — https://telegram.me/. You can add the parameters start or startgroup to this link, with values up to 64 characters long. For example:https://telegram.me/triviabot?startgroup=test
Telegram отдает данные через WebHook, но в json ответе нет нужных данных.
$field = !empty($_GET['start']) ? $_GET['start'] : 'значение не задано';