Gauges are notoriously inaccurate. It sounds like it’s working & we can check it with a laser temp gun, but I suspect it’s OK & if it overheats, it will let you know by pegging in the hot range & boiling over. Sometimes (me included) put a little too much coolant into a radiator & as it heats, it expands and may “puke” a little bit when you shut it off. Nothings moving (air thru the radiator & coolant) and engines always get hotter after you’ve shut them off after they’ve been running. Typically the car’s telling you that it had too much coolant to handle the expansion & it’s letting you know. If it continues to do this, you may have a bad cap or a faulty thermostat.