Other than just wanting to travel light, are you wanting to avoid (as much as you can) checking a bag period, or just avoid paying to do so?

Presumably you're coming from the US, so your longhaul flights will have at least one checked bag included for free per person (unless you're using a carrier not listed like Norwegian for the transatlantic portions). How many intra-Europe flights are you taking altogether?

I sympathize with the argument over how much to bring, I've gone through that before! Though you may want to consider allowing for souvenirs and things you may purchase along the way.

Look for a backpack or small soft bag that is expandable and one that has straps to help cinch it up after it's packed, that can save space.

I also am a fan of a travel vest at times. I have one that's much smaller and simpler than the one posted above - look more like a photographer's vest and was only about $20 on Amazon. But it holds lots of stuff. I initially got it when traveling on some carriers that enforce carry-on weight restrictions but can help save space as well and may eliminate the need for another bag.

I am always entitled to a free checked bag so it isn't about saving money. Sometimes on reentry from Europe the connection time is iffy so I go strictly carryon to avoid baggage claim. I can always put the purse inside the personal item if need be, but it never has. As far as hogging the overhead, both the personal item and the purse can fit under the seat separately or together. Please remember, not every flight is so full that overhead space is at a premium, especially not transatlantic off season.

It would likely be at a premium if everyone decided the straightforward "1 personal item + 1 carryon" rule didn't apply to them.
On Saturday, I was on a flight where boarding was delayed because an individual didn't think a laptop bag plus large briefcase plus rollaboard was a violation of the rule. It was.

All those airlines have exact measurements and weight on website for carry-on incl whether you can have a personal item and a bag and dimensions for each. Go to the websites again.
Easyjet is the most difficult: ONE piece of carry-on - purse, computer or whatever has to fit in the one bag that has to fit in the bag contraption (though no weight restriction).
Whatever you do, esp on Easyjet, book checked bags in advance - it is much cheaper. I just booked a DUS-GLA-LGW-TLS and GermanWings is only 8kilos carry on (unless booking through a partner airline, then partner airline rules apply though fares are more expensive) and bag dimensions are an inch or so smaller than BA and EZ. Also domestic flts (BA in UK, AF in France) can have additional restrictions based on fare class. And to think I once shipped a 4-poster single bed for free on AA (I was an employee but still... even I was amazed).

Also a so-called 21" or 22" carry-on is generally too big as handles and wheels are included in dimensions. I now travel with an 18". Anyway, lighter is easier on cobblestone streets and up and down steps esp through London Underground and Paris Metro with few escalators/elevators.

EasyJet will likely be very strict in enforcing its rules.

Be aware that KLM has a weight restriction on cabin luggage:
Your carry-on suitcase plus your "personal item" together may not weigh more than 12kg.
I do not know how much they enforce this rule.

I did have an experience with Austrian (I know you are not flying that). They weighed my carry-on at JFK. Their limit for the carry-on is 8kg (same for Alitalia, Turkish, and Lufthansa).

A suggestion: do not pile (hook it, attach it, etc.) your "personal item" on the top of your carry-on. It creates a visual impression of larger size/weight, and may result in a size/weight check.

piggy back on this. I talked to united . did u know your carry on can weigh 50 lbs. that's a lot of weight. but it would have to be rocks in order to fit it into size requirements

The TSA officer might be right in saying that you can take extra bags on board some airlines in the US, but European carriers can be quite strict in enforcing their rules. If you did take over the maximum allowance stipulated on their website they can insist that you check your additional bag in and if you complain, they'll just say that you should have checked the limits. Gate-checking the bag isn't free - Easyjet charges £45 per bag.

We've recently flown with Easyjet from LGW to INN. I took a backpack that was within their standard size and a handbag across my body. The flight was quite full, but I wasn't challenged on the handbag even though it was, strictly speaking, an additional item. I suspect that it was because it was a reasonable size (a square flat bag about 20cm wide), and not one of those big chunky ones. I don't think that they would allow a backpack in addition to the standard carry on. I was also prepared to stash the handbag inside the backpack if they'd insisted on enforcing the one piece rule.

When checking into another flight (BUD to RTM on Transavia) we had to put our bags into a metal sizer and anything didn't wouldn't fit would have had to be checked in.

If you do want to take an extra bag and check it in, do it online as BordeauxBlaise says, as it's much cheaper.