The cyrophorus approach looks, and acts, like a souped-up version of a Solar Still as used in camping. See part 4, Solar Stills, in this link: (not very detailed but a place to start)
https://www.wikihow.com/Get-Water-While-Camping#
The difference in your case is you already have the heat source (the hot water). What you seem to be searching for is a way to increase the efficient usage of using that heat to obtain distilled water.
Two straight forward approaches are:
1) use a heat sink of some sort that is colder than the water to condense the humidity. If a separate chamber or container is used, you have to ensure air/moisture circulation between the hot and cold parts.
2) use reduced air pressure to increase the evaporation rate from the hot water.
It sounds like you are trying to use both approaches, which would surely increase the collection rate. You still need to ensure moisture circulation between the hot and cold parts though. Perhaps a solar electric panel driving a small fan (computer fan?) would be adequate for a small to medium installation. I'm thinking to rather large passages between two chambers, one for hot-to-cold and the other for cold-to-hot.
The biggest problem may be maintaining a temperature difference though. Can the cold side be immersed in a local body of water, like a pond, lake or ocean? The advantage of that is their temperature remains close to the daily average temperature, not peak temperature. And don't forget a Sun shade for the cold side. Another cooling approach is evaporative cooling, keep a wet blanket wrapped around the cold side if nothing else is available.
I would suggest you try a prototype without the vacuum first to get all the details worked out. Remember that a good vacuum will easily collapse most common containers. Perhaps a beer keg, if you can find one, would withstand a mild to moderate vacuum. Those 20 and 55 gallon (80 and 200 litre) drums are not near strong enough.
Have fun! and let us know what you come up with.
Cheers,
Tom