Make of car for 4 adults

Good luck with that. 4 adults plus luggage is quite a load to haul around.

My advice in Ireland is smallest car possible. Last trip we were 3 adults plus wheelchair. Found a VW Golf to work well.

Be aware that when you rent a car, there's no guarantee that you will get the specific make and model you see pictured (in fact, IME, one never does). You will get something considered in the same "class". They will tell you how many people (and hopefully how many "bags") it can fit, but such descriptions may be a little optimistic. I would not pay much attention to the specific vehicle make and model they are displaying - it's only a general guideline.

You are going to run up against the competing parameters of comfort and drivability. Four people each with one large and one small bag will require a big vehicle. Ireland has lots of small country roads that seem impossibly narrow, and on those you want the smallest car possible. If you're only going to stick to motorways (big highways) that won't be a major issue. But if you're headed off to Dingle or other small places, the car you rent that's big enough for 4 adults and lots of bags is going to feel like you're driving a battleship. Be careful.

Unless you have paid a premium price to get a specific car, the typical car rental is a 'class' of car. I would be surprised if when you did the booking it did not say "or similar" by the car description.

On one trip ( US domestic) I had rented a 'mid size' car, "Chev Malibu or similar", and ended up with a Toyota Corolla. To the rental company they were the same 'class'. As it was jut two of us, it was not a problem, just a surprising revelation of how 'stretchy' a definition can be.

What size 'carry on' are you taking? A 20 liter back pack or a 20 inch roller bag? Big difference

I appreciate the feedback, but I am still in a dilemma on what car to rent. We will not just be driving on the main highways, but the back roads also. Has anyone rented the Opel Insignia?

The Opel Insignia (or its full size equivalent) looks like the best option in your particular case. There's a lot to be said for passenger comfort - especially for adults sitting in the back. You really don't want the trip to be an ordeal for anyone by trying to shoehorn everyone into a vehicle that's just too small for your purposes.
That said, the advice of the others regarding the narrowness of some of the rural roads (the L and R roads on the map) is correct. In many cases the roads neck down to only a single lane, requiring one of the opposing vehicles to give way by pulling off onto one of the frequent pullouts that are there for just that purpose. I guarantee that your heart will be in your throat the first time you meet a truck or a bus coming in the other direction, but the Irish drivers deal with this situation all the time. Though it takes some getting used to it's really no big deal. I do recommend using the street view feature on Google Maps to preview stretches of road you plan to drive just to familiarize yourself with the sight picture. It helps that the local drivers are some of the of the kindest and most patient you'll find anywhere. Our experience has been that in the time it took me to recognize the situation, slow down, and start looking for a place to bail out, the opposing driver had already pulled over and was waving me thru.
The default rental in Ireland is a manual - are you ready for that? In most cases it helps a lot to specify an automatic at booking. It'll cost more but it's worth it. You just don't need the extra fun factor of trying to manually shift with your left hand while you're trying to deal with everything else.

Has anyone rented the Opel Insignia?

If you really want to get an idea for how much space the car should have, go to your local Buick dealer, and look at a Buick Regal - that is apparently the same car (according to wikipedia). But...Let me be blunt: stop obsessing over a specific car model. You are NOT going to get that. You will get a car that the agency believes is "similar" - their definition of "similar" being the same number of seats, and approximate cargo capacity, engine size, etc., but it will not be an Opel Insignia. It could be any of a hundred cars in the same "class."

They give you the car that they have available that day. Renting a car is always a roll of the dice.

"But if you're headed off to Dingle or other small places, the car you rent that's big enough for 4 adults and lots of bags is going to feel like you're driving a battleship. Be careful."

Fear not. Five years ago we walked over to the rental agency near our Dublin lodging to pick up (we thought) a small VW sedan-Polo or Golf. As promised correctly above, they did not have one. After some discussion we elected to take the Nissan Qashqai as an alternate, for the same price. While there were one or two times I wished it were a bit smaller, it was just fine, nothing at all like a battleship. The Qashqai, currently on offer from Enterprise at Dublin Airport and possibly others, is classified a "small SUV" with room for 5 adults and "3" bags. I will presume that your 22" bags are smaller than the tested bags...

If you are average sized folks, I do believe this vehicle will work for you, I hope that's helpful. As well, if they did not have availability when you show up, then in theory you'd get a vehicle of the same class. It would also be prudent to look at the Nissan X-Trail, for example-that's the "medium SUV" from Enterprise-in the event you get "traded up". I see that Auto Europe also shows the Qashqai.

A search re luggage capacity brought this:

"Nissan Qashqai Luggage Capacity
The Nissan Qashqai, though a compact SUV, has an exceptionally large cargo capacity. Able to hold up to 4 suitcases, the 430-liter trunk ranks among the top of the CUV class. Families with lots of luggage would do well to consider renting this versatile vehicle."

Note that "Full Cover" insurance is generally the way to go in Ireland.

Thank you all for your helpful insight on car rental. I will leave my Opel Insignia reservation and hopefully it will work out. Thanks Again!