There is a problem with the live streams in Safari. If the player is paused for a period of time during a live stream and then resumes playback, the player does not jump to the current position. The player often plays from the buffer first and only then jumps to the current position.
It sounds the expected behaviour. When the player resumes playback it will do from the point where the playback was paused, meaning that it will play the data from the buffer first and then continues downloading new segments as they are required. Take into account that the playback will be time shifted. Can you provide a more detailed explanation about the use case you wan to achieve?