I've been a DM for a while now and I've only ever had this problem with my larger groups (6+ players). They never try to hide this "habit", and they "admit" to it, because they think it's perfectly fine and that I'm the one placing strict rules on them. So what am I talking about?

My players play video games and do other things during combat. When I ask why they aren't paying attention, they say that they have no right to listen if it's not their turn in combat. They only really focus when it's their turn. This isn't to say they're not devoted players - they all are - they just don't see any point in focusing if it's not their turn in combat. If the monster they are fighting is killed, it takes me at least 15 seconds to get everybody's attention and I often have to raise my voice just to ask somebody what they are doing on their turn. Sometimes they are literally watching TV shows or playing online chess whilst I am speaking and directing combat, and when I ask them to recap what has happened, they say "I don't know." 

I've tried many solutions to this - setting stricter ground rules, "banning" online games, and even switching to paper character sheets. I felt those were too strict. As of now, the only rule I have is that I will occasionally ask people to summarise what has happened in the past few minutes. Is this okay? What else should I do?

Have you tried guilt? It works wonders. 
“Folks, I spend many hours every week getting this game ready. The least you can do is to show some respect for my time and effort and pay attention.” Or also, “Damn. Get an attention span. I know goldfish who can focus better.”

The idea they have no right to listen when other people people are going is absurd. It’s a team game. They should be working together and suggesting strategies. “Can you move back 5’ so I can fireball on my turn?” or “I’ll knock this one prone so melee types can get advantage.” Not to mention knowing when to use their reaction. I don’t see how they can function without paying attention to other players.

You also might check in about if they actually want to play, because it sounds like they don’t. They may say they want to, but their actions say otherwise. If that’s the case, find out why. Could be tabletop isn’t for them. Could be they are not finding the story interesting, and you need to shift up your style a bit.

In their defense, I will say, 6 players is a lot. When you add in the DM, a seventh player, it can easily be 15 minutes between turns, sometimes more. And none of them paying attention makes turns longer, so it makes more time between turns, so it’s probably making the situation worse.

People think combat is boring when they don't focus, don't preserve the momentum of the scene, and aren't invested. The problem isn't combat, it's the attitude casual players approach it with. 

Ask your players to bear with you and try something: everyone pays attention, plan your actions based on what's going on, don't take more than a minute to execute your turn, make active choices, etc.

Insist on it the next few times you combat and see if they get into it

If you feel your players actually respect you or can be mature, I'd just bring it up. "Guys, this is a collaborative game and you're being rude to me and your fellow players. It isn't fun for me when you don't pay attention." 

You might also try sides initiative, or group skill challenges to speed up the pace and keep everyone on their toes at the same time. If they respond positively to faster play, maybe it's just a matter of adjusting. If it doesn't change anything, maybe that group isn't meant to be. 

Honestly, if it was me, I’d just come out and say, guys I have to put a lot of time and energy in to planning and building these sessions and encounters, the least you can do is pay attention, if you can’t at the bare minimum do that, why are you even here?

This idea that they shouldn’t know what is going on is nonsense. While there is a lot of planning and discussion that happens in a a battle in D&D that wouldn’t in real life, even in a real life fight, combatants are paying attention to what is happening around them and what is happening to their team. If they want to play it strictly realistic and say they can’t coach each other or discuss things in the heat of battle that’s maybe reasonable, but they would still know and watch what is happening around them.

Our house rule was loosely one minute to declare actions for your turn.  We were never strict about it because any time overruns weren't because people weren't paying attention but because they weren't completely sure how something worked or the battlefield had changed significantly just before their turn.

It never really came up, but the rule was that if you weren't paying attention, you dodged.

If they're low enough level an unexpected character death might sharpen someone's focus.  Jeff, sitting there playing Candy Crush is told "You suffer 22 points of slashing damage and another 18 points of fire damage from the first hit."  Poor ol' Jeff looks up in shock and sees his 32 HP are gone.  "The second attack is a crit because you're down."  Works even better if the next enemy is near him and Thundersteps away.  Jeff will need a break from Candy Crush to roll a new character while we resolve this fight.

There's 4 in the group I play in and the group I run and combat can sometimes bog down a bit.  Minor drifting is overlooked but prolonged inattention is targeted and I find letting someone know they got hit for half their HP brings their attention to bear pretty quick.

I've seen this in a couple of groups and had limited success dealing with it.

One group had several players on their devices, and their excuse was always "it takes too long so I'm doing other stuff." They didnt take it well when I pointed out that the reason that combat took so long was that I had to repeat myself six times for everything I said. I started refusing to repeat myself and not answer questions, but that just led to bad blood, so I backed off. Also, the time saved by not repeating myself was more than taken up by players whinging.

Not sure what a final answer is. I suspect it does involve just being firm and shutting down the whinging, and accepting the bad blood.

I do enforce "no takesbacksies" though. If I roll to hit and a player is not paying attention then I roll damage and tell them the number. If they go, "Oh, I shield/parry/whatever" then I am firm in saying, 'too late for that. 15 damage, mark it down."

One thing I do which seems to help keep players engaged (though this is over the table rather than over the internet so they will have a board rubber thrown at them if I see them doomscrolling) is to tell the other players what they see when a character does something.

So, rather than:

Bert: "I swing my sword, that's a 19, hits? Ok, roll damage, 9."
DM: "You swing your sword and strike sparks off their armour, leaving a glowing rent in the steel"

I would go for:

Bert: "I swing my sword, that's a 19, hits? Ok, roll damage, 9."
DM: "Azreal, standing where you are you see the sparks fly from the monsters armour, and even from this distance you can see the strange glow coming from inside the rent steel."

Thus Azreals player is engaged on Berts player's turn.

I don't do it for everything, and often will use it to engage players whose characters are out of position (the barbarian is never in run & rage distance at the start of combat. Ever!). It is particularly effective when someone wants to do something risky or awesome, and even more so when there is a pause available. A loose example of this was in one of my games. For context, the warlock was riding atop a rampaging rhino inside a circus tent, and the paladin was getting his butt whupped by an Ettin artificer in the wooden building in the middle. She attempted to steer the rhino and rolled a 15 on her Strength check.

I could have said "You feel the beast start to turn as it swings into the building you were aiming for", but that gives the game away. Instead, I said:

"(Paladin), you see the Ettin laugh at how little damage your blows have done to his armour. He raises his sword to strike you, and the entire room seems to lift for a second - the swooping sensation accompanied by a splintering of wood and the roaring of a grat beast as the floor gives way beneath the Ettin, pitching him down to the ground below"

I then returned to the Warlock and say "can I get a Dex check to either stay on the rhino or jump off into the building you just successfully demolished?".

It makes for a slightly more cryptic storytelling, and I can see my players pay attention as they never know which of them I will talk to - I never ignore someone until it's their turn.

Another mechanic is the slightly comedic calls for saves. In this example, I might have the Barbarian who was to the side of the building make a dex save, and if they fail they get a piece of wood hit them in the stomach from the splintering building - no damage, just flavour - splinters in their beard, that sort of thing.