"I now realize the value of multiple catalogues."
That usually is a bad idea that leads to confusion. You remember an image and want it. Where do you look? The common exception is the wedding photographer that shoots a thousand plus and wants to keeps the Jones separate from the Smiths.

"Can an image be worked in each program?"
I does not seem to be accepted much on UHH, but Adobe delivers a system of linked parts on the $10/month plan. A photo can be "in" Lightroom CC, Lightroom Classic, Lightroom Mobile (phone), Lightroom Web, Photoshop and a private Portfolio. That photo is linked or "synced" so that an adjustment anywhere shows up everywhere.

For example.... You shoot a masterpiece on your DSLR. You are working it to perfection in Lightroom Classic and maybe Photoshop. Your granddaughter invites you to lunch and says, "What have you been doing lately Grandpa?". You whip out your phone and show her. She says it is too dark and needs some cropping. You adjust it on your phone, in the restaurant. By the time you get home to your desktop machine, the corrected image is there. It beats you home!

Or, you're with your buddies at the local pub discussing the last dirt bike adventure. Your iPad, Android tablet or laptop are in the car. The buddies need to see the pictures you've been developing on your desktop. You already put the good ones in a Classic collection. Lightroom CC "knows" that so you can whip out your "device" of choice and show them.

It works the other way too. You might be at Costco and see a DJI drone. Grab your phone, take a snapshot of the price and details of what is in the kit. When you get home, that snapshot is in Classic. You can compare the Costco package details with the Fly More Combo offered at B&H. You discover Costco is cheaper because it has two batteries instead of three! Then you can go back to Costco later in the day and get the better deal. (A friend told me that story!!!)

Lightroom 6.14 was fantastic for all that it did. The new Lightroom/Photoshop system is far ahead of it. There are new things in Classic, like masking. But the collection of the system parts go even well beyond that.

There are basically two Lightrooms, one on line just called Lightroom and the other Lightroom Classic, both are $9.99 a month. The first is totally online product, which means you can edit your pictures anywhere that you have an internet connection and all your pictures are stored in the cloud. You get 1TB of storage and there is no access to Photoshop for the moment. The second option which is what most in this forum who use Lightroom have is Lightroom Classic subsriptions which include Lightroom online and on the desktop. This also has access to Photoshop. The other difference is that you only get 20GBs of storage online. You can upgrade to $19.99 a month for 1TB. LRC is far more powerful than LR Cloud and since you are already using Lightroom 6, the only learning curve is the new advancements that have been made to the software that you already have. So bottom line, I rcommend LRC for you.

There are a lot of other benefits if you decide to use the online version that comes with the LCR subscription. You can upload your pictures to the web via a catalogue in LRC to the web for viewing anywhere and you can share them as well. As long as the pictures remain on your hard drive, they do not count towards you 20GBs. You can also create a website for free. If you install the Lightroom Camera on your phone, you can set it up to down the load the pictures automatically to your computer hard drive, even if you use your native camera software. Once that is done you can delete them from online so they do not fill up your storage. Lightroom is a whole photography eco system.

You said that you only do light work in LR6. You may want to look into Photoshop Elements. I saw it on sale at B&H a day or two ago for a on time payment of $59.99. You could also do a trial run of LRC and see if you like it. If you are using the organizer, (catalogue) in LR6, taking the subscription would probably be your easiest move. The major change would be your access an expanded number of benefits by upgrading.