For 500mS delay, you need to calculate capacitor voltage after one RC time constant, which should be 37% of Vcc (full charge capacitor voltage), according to formulae Vc = Vcc X e
-t/RC..
Of course, you can set any RC delay point (e.g. 0.7RC for 50% discharged capacitor) according to convenient R & C items values..
(500mS time constant can be set either by 500k & 1μF or 500R & 1000μF or any other combination of 10
x X 10
-x to set units for sec..)
BUT, in your first case, R can't be considered relay's resistance 'cause 1N4007 flyback diode is set in parallel with relay winding..
(and I shouldn't count on diode's incremental resistance for setting RC constant..)

For BJT drive, set time constant (4XRC) for about 20% cap charge (80% discharged), if you trust on 1.2V release voltage..!!
(for more precise delay setting, measure particular relay contact release voltage and take into account electro cap tolerance..)
Of secondary significance is BJT relay driver for longer life mechanical switch contacts due to lesser current through them..

edit: Not enough coffee yet..

You can always implement first version without flyback diode and count on about 50R coil resistance..
LED may suffer from reverse voltage kick but we can't always get what we want..