Mine is a 951. The problem I have heard w/ the 561 relates to its losing the ability to come on at all, not just when the key is on and the engine is off. The ground from the relay runs into the ECU. When the ECU detects a signal from the coil, it completes the ground and the fuel pump runs. The common (but dangerous) work around to this failure of the 561 is to tap into the ground from the relay to the ECU and ground it to the body so that the fuel pump runs whenever the key is on. If you ever get into a wreck and the engine shuts off, the fuel pump will go on pumping fuel - not a good thing if your fuel line gets cut.
This is not your problem, since your pump obviously runs when the ECU gets a coil signal. I think you have another failure in the ECU, but I'd have to look at the pin out to be sure. I suggest you get ahold of a good pin out diagram.

Now that you mention the reason, I remember a discussion of this vapor lock. But unlike you I hadn't discovered this for myself so I suppose it went in one ear and out the other. Thanks, Phil.

Maybe I can apply this to my 79. I've been thinking about checking my rest pressure when it gets warm again, because it seems to crank too long after sitting awhile. Not real bad, but at least twice it felt like it wasn't going to start again just after refueling immediately after running at 4000 rpm for 50 miles. But most times it starts fine, so I haven't had the motivation to pursue this in the snow.
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Art Benstein near Baltimore

I've heard this too, about the ECU priming. I know none of my Volvos do this, but I've heard many domestic FI cars buzz a second when you turn on the key. It does wonders for the starter's longevity, I think. Sounds like it might be a good mod too if done correctly and safely.

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Art Benstein near Baltimore