Thank you all for your help and ideas. The comparison tables are also very helpful.
As was delicately pointed out by Pr2, though, and also by pop_eye, the Prismcube is not an option for me unfortunately, regardless how wholeheartedly comitted its maker may be to open source.
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Regarding Coolstream, I found this German thread from beginning of 2012, presenting the then brand new "TwinTuner" Neo² (which is the only Coolstream box I have considered, with twin DVB-C, of course). At least some of the posters are, I believe, also developers of Neutrino HD. There it is said that simultaneous recording on both tuners is possible, and also, of course, watching on one while recording on the other. Furthermore, if those reports are to be believed, the user only has to concern himself with the selection of a channel, and the system will automatically change tuner as needed.
Quoting PauleFoul in that thread:
Im normalen Betrieb der Box merkt man gar nicht das man einen Twin-Receiver benutzt, nur ein kleines Symbol im Infobar beim Umschalten zeigt den verwendeten Tuner an. Man muss schon sehr genau hinschauen damit es überhaupt ins Auge fällt.
(Which I think translates to something like: "Under normal operation of the box, you'll not notice you're using a twin receiver. Only a small symbol in the info bar when switching channels will show what tuner is used. You have to look closely to notice it at all.")
Also, Dutch Astrasat (now apparently the main European distributor of Coolstream) has a page on the Neo². (I don't think I can link to it under the forum rules since they seem to be selling the box directly, but you can probably find it for yourselves.) There they claim (in Dutch, but there is Google translate) that this box allows you to record no less than 7 channels and watch an 8th (!?!). At first I thought they must have been smoking something, but it seems it's common for DVB boxes to allow several channels per tuner to be simultaneously recorded (or watched). Again, I am a newcomer to this.
Finally, Tuxbox.org (the ones who seem to have started it all) has a wiki for Neutrino HD (German only, unfortunately). Where it is said that a warning symbol will be shown for any recording that overlaps with a previously set recording, but that this is relevant only when no more than a single tuner is available and the recordings are on different transponders. Which to me seems to imply that, when two tuners are available, both can indeed be simultaneously used for recordings, and that Neutrino HD will automatically make this happen.
(On the same page, it's also said that, with one tuner, PiP is only possible with pictures from the same transponder, but, with two tuners, Neutrino HD can show PiP from any two different transponders.)
I have to believe you, of course, when you tell me that Enigma2 deals more intelligently with tuner selection than does Neutrino HD (you clearly have an understanding of the inner workings of these systems), but couldn't it be that the AzBox in the YouTube video was configured in some non-standard way? I see that tuner use on the Coolstream Neo² can be configured in a number of ways, one of which is "loop" (which would be the normal use of twin DVB-C tuners), another being "independent" (only for sat tuners).
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Conax cards can run on 5MHz, being a class A (5V) card, it should support frequencies in the range of 1-5MHz according to ISO7816-3.
So you should not worry about destroying your card.
Thank you. This sounds very reassuring. I was indeed afraid the Xtrend card reader might even destroy the card. Seems I don't have to worry about that at least.
The only doubt then, would be whether it runs well (no software crash).