After this first start, I was driving about 3 or 4 km's and on my second start, to get back home again, everything worked perfectly again.
On my way back home, which usually would have taken me only a few minutes, I was driving the car for about half an hour to charge the battery again.
Tomorrow, on my next trip, I'll see, if it's ok now.
I plan to give the battery another half an hour or a bit more of driving, to make sure, it gets charged as much as can be.
As for a new battery, I wonder, why everybody at my location (not only my Honda dealer, but several car accessory dealers, too) keeps telling me, there is no other battery for the Civ, but here on Civinfo, there are lots of postings telling me the opposite?
Are all those guys just to lazy to replace it?
Unfortunately, there's no electricity in my garage, so driving is the only chance of charging.
Like I said above @ Woette, I'll do that tomorrow, because, when I think about it, I realize, my last few trips where too short, probably.
Anyway, it's very annoying, if a car keeps telling me, when or how far I have to drive it. I once thought, that's up to me to decide.
Well, despite the fact that the Civ certainly is a beauty, I guess, my next car won't be such a "driving computer" with tons of electronics build in, anymore.