Alex,
No you don't understand the meaning of Modulo. Modulo is used for numbering systems.
Some examples:- binary is Modulo 2 (it has only 2 digits 0 & 1), Modulo 10 is the decimal system (digits 0 to 9), Modulo 16 is the Hexadecimal system (digits 0 to F).
The circuit you have drawn is a Modulo 10. It will start at 6 and count down to 0 and then continue counting down 9, 8, 7 etc since it is a Modulo 10 counter, not a Modulo 6.
To make a Modulo 6 down counter, you have to design it so that when the count reaches 0, it jumps to 5 at the next clock pulse. So it counts 5, 4, 3, 2, 1, 0, 5 ,4, 3 etc.
See the attachment below. This is a Modulo 6 down counter
Why do you want to use relays? CMOS logic would be more energy efficient and require less space.
If you want to use relays, you can make some savings. Relays R0 and R1 are not necessary. the "Zero" signal can drive R3 directly and the "One" signal can drive R2 directly. When I say directly, I mean it in a logical sense. You will need 2 transistors to physically drive the relays.
The Green LED will be on continuously until R2 operates. You wanted it to light when the Start button was pressed.
If R2 operates but the circuit is reset before R3 operates, R2 will remain operated.
Len