There are many rivers that flow through many lakes and keep their name all the way. Therefore, waterway=river is a non-false tag in this case. For example: Relation: Кемь (9445490) | OpenStreetMap
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I would leave this up to local interpretation. Certainly where an area of water is called both a river and a lake by the locals I would consider it correct to tag the area as a lake and a line though it as a river (I’ll edit my post to acknowledge this). There are plenty of other places where the locals don’t consider a certain lake to also be a river, though. In this case it is false to tag a line going through the lake as a river. This line for example does not represent a river. It is only considered a lake.
It feels like we are overcomplicating things.
If a river flows through a lake, then draw it through the lake.
If a river flows into a lake, terminate it on the inlet.
If a river flows out of a lake, start it on the outlet.
Whether the river goes into, out of, or through a lake is largely a matter of convention and nomenclature, and that’s OK. After all, that’s what we’re mapping – nomenclature and convention.
I don’t buy the argument that “connecting the dots” between inlet and outlet is some kind of massively hard problem. If a child can do it with crayons, I’m pretty sure an algorithm can figure it out. And, if the end-user application only needs to associate inlets and outlets together (and not necessarily connect them with drawn lines), this problem is even easier.
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There are many many things that children can do, that computers cannot. e.g. A child can walk on 2 legs, balance, and avoid obstacles. Computers & algorithms barely can do that.
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Not sure if it’s based on OSM, but it might as well be:
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Hmmm, I’m not sure if that’s an argument for or against! But it’s neat to see in any case. 
I have so many questions!
1) What is the use of waterways.org beyond pretty visualisation of streams and rivers?
2) What is the reason why it does not assume rivers are connected by bodies of water? Law of physics kind of suggest that if one had inflows into a lake/reservoir/, the water from them eventually flows to outflows. These are given by directionality and waterways/natural=water should be connected by a node. Having read this thread and the wiki proposal, I do not get what this is for at all.
Looking at some rivers mapped as multipolygon relations, I can see that it is common practice to map large rivers as areas and then add a waterway=river through them as per Relation:waterway - OpenStreetMap Wiki and Rivers - OpenStreetMap Wiki. However, both wiki entries are silent on what to do when river enter lakes or reservoirs. For example the Vltava river Relation: Vltava (1730536) | OpenStreetMap is dammed most of its way. There is a waterway=river running both through its unregulated flowing parts and the dams on it. 3) This proposal would replace the latter waterway=river into waterway=flow, right? Such a change makes sense to me, I guess, if mapping rivers through stale bodies of water is accepted practice. 4) But is it?
5) And is there a rule against putting areas into relations?
To me personally, when I search in an app a river with dams, I would at most levels of zoom that are not above (resonably-sized) country level expect the river being highlighted as a line where it is unregulated but as an area where it is not. So it would make more sense to me not to map anything through natural=water and let the consumers handle it based on intrinsic logic. However, if it is accepted practice to map through lakes and dams (could that be then documented please?), then flowline seems to be better.
Also for the proposal itself, I think it should at least generalize to also include reservoirs but also moats and basins etc. perhaps:
Where a watercourse tagged as waterway= enters a body of water tagged as natural=water and ceases to be recognizable as a stream or river, start a way tagged waterway=flowline.
However I guess I prefer the alternative proposal if I had to choose.
Also, this post should nto be forgotten:
Based on that, adding to proposal something like. “When a waterway enters a natural=wetland, tag with waterway=flowline and interminent=yes.”