I will make it simple for you. Vents are either ducted outside or they are not. Internal can have to do with the location of the blower.

so are you saying that a Ductless hood is the same thing as recirculation hood?.... and that the hoods I selected require a recirculation kit?

Yes, ductless and recirculating hoods are the same. Internal just means the blower is built into the hood. Some hoods that are vented out can have the blower mounted separately, but that does not concern you.

All 3 of the hoods you linked come from the factory set up to vent out. Your first link (Broan) gives the part number for a recirculating hood, which you would need to buy if you went with that hood. By the way I own that hood, and really like it. Mine is vented out though. Your second link (NXR) and third link (Kobe), make no mention of a recirculating kit, so I can only assume they are meant for venting out only and thus would not work for you. Note that I did not dig deeper past the web pages you linked, so there may be mentions of recirculating kits elsewhere.

What range is this going to be going over? The professional style hoods you linked usually go over high power professional ranges. Owners of these ranges usually vent out, because of the more intense cooking they usually do and the higher heat they put out. Because of this there will be fewer "Pro Style" hoods that can recirculate vs. other types.

I have a new ge cafe duel fuel converted to propane. I rarely will use more than one burner at a time.

One thing that I haven't seen mentioned in this thread yet, is that the phrase "recirculating kit" could refer to a couple of different things.

Some range hoods can be ducted outside or set up as recirculating "out of the box". If you want to duct it outside, you only need to buy the actual duct (from Home Depot > Ventilation aisle). If you want it recirculating, you have to buy the appropriate charcoal filters ("carbon filters" - from the hood manufacturer).

These hoods will typically have a series of slots or holes at the top of the chimney, so in a recirculating installation, the air can come back out into the room. Here's an example:

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In addition to the slots on top, there's also a provision to mount the charcoal filters somewhere on the bottom of the hood.

In the case of a hood like this, the "recirculating kit" is only the charcoal filters, typically in the $ 20-70 range.

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However, some hoods do NOT have a provision to be recirculating, unless you buy a special piece that will let the air come out (the slotted piece above the hood in this picture):

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...and THEN you can buy the charcoal filters. So in the case of the Viking "recirculation kit" above, there's a $ 390 upcharge to make the hood -capable- of recirculation, and -then- it's $ 79 for the filter.

So, be sure to find out what the phrases "recirculating kit" or "capable of recirculation" or "ducted/ductless option" refer to - different manufacturers will have different interpretations.