I just implemented the latest modification to the circuit (moved the capacitor across R2). But now the the relais already reacts on the last 2 of 5 pulses.

The timing should not have changed.
Is the capacitor polarity correct as shown in post #19?

 

It could be that the circuit is operating marginally.

Try one of the following.
1) Change R1 to 1.2kΩ
2) Change R2 to 680Ω

An alternative is to go back to the original circuit. The high charging current only happens on power up.