If the jars are refrigerated they're perfectly safe. Actually, since they're jams, they're probably fine on the shelf too. With the bigger headspace and the short processing time there's a weaker seal and more risk of oxidation of the surface but it's not a food safety issue in the case of preserves.

Rings are removed for two reasons. One is just that a lot of people buy boxes of flats but re-use the rings from batch to batch. The other is if there's some seepage during processing, the rings get "glued" to the jar over time and then they get difficult to remove. They also corrode. But if you want to leave the rings on the jars, feel free. I often use rings with jars I gift. They look nicer.

Carol

I always leave my rings on. I do not tighten after them after they come out of the water bath. The rings are more likely to rust which is the reason for taking them off. I've only had to buy rings every few years because of them rusting.

I often do exactly what you did when I don't have quite a full jar. Can it. Hope it seals and stick it in the frig as soon as it's cool enough. It should be fine.

Yes, thats the main reason they are removed, if they have been in a water bath and there is residual water left under the rings. Here, my steam canner seems to scrub that area of any corrosives so the rings don't usually atart rusting unless the contents has been opened, is a vinegar salt brine that has leaked out, and is in the fridge after opening. Thats one reason Ball started to offer plastic caps. These are NOT leakproof, but offer no corrosion once the ring is removed and replaced with the plastic caps. I use both the lid and plastic cap to keep a jar sealed after opening and is stored in the fridge.

Rust isn't my main reason for removing rings although it sure helps prevent it and the damage to the jar threads than can result.

My main concern is the molds, yeasts, and bacteria that can grow there in that nice dark, snug, place from the boil-over during processing and any residual dampness. I won't go into the exploding jars that can't pop their lids because they have a ring left on. ;)

Per the BBB - "Before placing the jars in your pantry, remove the bands and wipe off the lids and jars to remove any food residue. Rinse and dry. Do not replace the bands. "

Since it is recommended that the jars and threads are all wiped down before putting in storage anyway, the rings have to be removed to do that. So why wipe down the jar and threads and then put a dirty ring back on? And, it also gives you yet another opportunity to check your seals and discover any weak ones.

Once the jar is opened for use it is easy to simply add a ring back on the jar or use one of the plastic ones for fridge storage.

At any rate, it's up to you of course but I have linked two of the longer previous discussions on this that you might want to review. ;)

Dave

Rusting rings on jars

Here is a link that might be useful: Rings on jars?

I wash and dry the bands after use and store them by size in two covered plastic containers with handles. It's also a good time to inspect and discard any with condition issues.

Carol

After removing the rings, I place the jars in my sink with a few inches of soapy water and use my vegetable brush to wash the threads, Rinse the jar and wipe dry. Lids are washed and dried thoroughly and stored is a sealed container as Carol said.
If I am giving something away, I replace the rings and add a few of the plastic replacements.
Ball now makes a replacement for the wide mouth.