the parks kind of have some signs when it's really crowded or they are expecting it to get really crowded.. I'm just going to focus on MK as it's similar in all parks. they also appear to happen in this order..
1: more security then your used to seeing and they aren't ghosting each other (training appears to have a period of time where 2 security guards work together and 1 is doing more overseeing of the other). If it's just more security available to check you, the park is either your going to get nothing done crowded or it's expected to turn into that before the shift ends.
2: more counter service locations are opened. I look across to the one that is in with the desert party or the one that is across from the pirates ride. those are usually closed unless they are expecting more capacity then the usual always opened ones are getting.
3: cast members start to hold signs in narrow walkways telling you to stay right. commonly found in the back to help people walk around between the carousel all the way to liberty square.
4: the line of castmembers is more of a wall of castmembers and they are verbally louder in staying right and calling out people not staying right to get right.
Meanwhile..
A: rides that generally are short wait start to have 20-30 minute waits and the popular rides start to break an hour.
B: all rides are long waits.
c: peopel hang out for hours in the hub for the best spots.
Now that all happens without any type of phased closing. July 3rd and 4th or any major holiday, magic kingdom will be the experience of all of them, however it doesn't really start to get heavily crowded to the point you start to give up on rides until closer to dinner when everyone just coming for fireworks start to pack the place. If it's a place where drinking is common with the celebration, then expect epcot to also be crazy full. If there is a special fireworks or show for the holiday in the park it's disney trying to get people to spread out to many parks and everyone is crazy... daytime holidays are more common to be more crowded near lunch like xmas when people think they are going to be in the live parade that isn't live.
The holidays with special fireworks at MK (july 4th and new years eve), it's an amazing time to rope drop, get out of there by early afternoon and then plan to catch the fireworks from a monorail resort. Those fireworks displays are the largest you'll find with only a couple exceptions of some shows elsewhere in the world being larger. They even close the boats down for about an hour to bring out barges loaded up with extra fireworks. I have some friends that love celebrating those holidays by booking a room at poly and just watching from there never setting foot in the parks at all. (check
youtube to see the fireworks, they are great and to be honest, in the park might actually be the worst place to see them).