two water pump project for newbie

Test_1a:
Connect the AC-DC 6V power to the pump (+ to +, - to -)
+RESULT: it works normal

Test_1b:
Connect the AC-DC 6V power , the pump and Arduino with computer USB powered (shown as diagram)

const int pump = 7;

void setup(){
  pinMode(pump,OUTPUT);
}

void loop(){
  digitalWrite(pump,LOW);
  delay(3000);
  digitalWrite(pump,HIGH);
  delay(6000);
}

+RESULT: the LED indicator on the relay input#1 works with the code, but the pump has no response at all.

Am i wiring wrong?

or the 12V relay can't work with 6V DC power input?
but if i turn the DC power to 12V, and digitalWrite(pump, 128) seems ~= 6v?