Hi Carla, and welcome to the forum:
At this stage I am assuming you do not want to pay for native RF long telephoto lenses, given their high cost and that's why you are looking at extenders, which are themselves not cheap.
I have used the EF 70-200 MkII IS USM (f/2.8 and f/4 units) with the R6, but not with an extender. Essentially, I see several questions one might pose here:
1. Should one use a RF extender with the EF-RF adapter attached to that, and then the lens, or
Use the EF-RF adapter, then an EF extender and the lens?
To be honest, I cannot answer that, but prefer to avoid using extenders as a general rule.
2. In either case which extender - 1.4x or 2.0x would be best?
I have used the 1.4x and 2.0x (both MkIII for EF) adapters with the EF 70-200 IS USM MkII (both f/2.8 and f/4.0 version) lenses and would honestly not recommend the 2.0x. You will lose two stops of light, and the optics are not that great.
3. Would it be more efficient to find a different lens instead of using an extender?
This is a possibility worth exploring, but much depends on your budget, which we don't know. Given the price of adapters is rather expensive, it might be worthwhile looking for another lens with a longer focal length range. I am assuming at this stage that you want the flexibility of using zooms, so the RF 600mm (for example), might not suit. Given that, and the high cost of RF zoom glass, plus the fact you have an EF-RF adapter, you could go for a longer EF zoom lens. You could go for the excellent EF 70-300 IS USM MkII lens - lightning-fast focus, good optics and stabilization. This will give you a 1.5 boost at the long end without losing the 70mm at the short end. Another possibility would be the EF 100-400 MKII L lens - giving you great optics with double the reach. You might be able to pick one of these up as people move to the RF 100-500L for their R-series bodies.
If you really want to go with an extender, I suggest you look to borrow or rent the options you prefer and see how that works out.
To answer your question, I have a 1D X Mark III which is the same sensor as your R6. I also have a 70-200 f/2.8L II along with Canon Mark III 1.4X and 2.0X extenders.
Do you have the extenders already? What is the end use of your images? If you aren't looking at making large prints try some cropping before you start spending money on new gear.
I have run some tests with my gear. I would not recommend the 2.0 extender; I can get equal or better quality, depending on the images, with the 1.4X extender and cropping. And I don't get the focus speed hit and light loss due to reduced aperture of the 2X. The image quality with the straight lens cropped to 2X coverage is extremely close with the use of some of the new AI software from Topaz and DxO.
What extenders are you referring to please? Are you using these on an R-series body? There are RF extenders intended for the R-series bodies and EF extenders for the legacy EF lenses - and since either could be used in different ways with the EF lens on an RF body knowing precisely what is being used could be significant and very helpful.
Since the Carla is using legacy EF lenses, she has an option to go for either.
Well, actually one could theoretically put an RF extender between the EF-RF adapter and the camera, hence my question. Also, there is a 100-400 RF lens, and it will use the RF extenders. see This link - in fact it does not have the issue of the RF 100-500 that it limits the use from 300-500 when attached.
You mention using the EF 70-200mm f/2.8L IS USM "II" lens... That works well with the EF "III" Extenders. It might also work well with the EF "II" Extenders, but is not something I've ever tried. (I have use the EF 1.4X II on the EF 100-400mm II with excellent results.)
I have no idea how that lens would work combined with either of the RF Extenders.
I can tell you that I avoided using EF 1.4X II or 2X II on the earlier EF 70-200mm f/2.8 IS USM (original version). Image quality just fell off too much. Especially with the 2X.The 1.4X II might have been acceptable to some, but I had two 300mm available, so no need for that combination.
Generally speaking, I avoid using Extenders as much as possible and use the 1.4X far more often than 2X. The stronger 2X has noticeably greater effect on image quality, so I use it very selectively... never on an zoom. But I have seen some good results combining the EF 70-200mm f/2.8 II with the EF 2X III, in particular.
I have similar experiences. I have both the 1.4x and 2.0x MkIII extenders - earlier ones are much more limited in how they engage with cameras' autofocus and aperture settings. I was using them with the 70-200 f/2.8 IS USM MkII and the 100-400 IS USM MkII zooms and found that the 2.0 extender degraded the results unacceptably - I was using them with Cano 5D IV, DsR, 7DII and 90D bodies. While I have had no issues using the same lenses on the R5 and R6 bodies I have, I have not considered using them with extenders.