The answer for peak shaving and ac in depends on whether you have a grid meter or not and how you have set up your battery.
The grid input limit is a continuous limit (never to be exceeded), peak shaving is short term ‘spike load support’. You can set this lower than the grid limit. The grid input would then be the fall back if battery limits are met.
(The limits apply to both - ac in only works if you have a grid meter). And it works well if the grid meter is not slow.