The coil and capacitor must be resonant at 125KHz.
If you change the dimensions of the coil, you change the inductance; a larger coil will need less turns to give the same inductance.
The other problem is simply the power level; the circuit drives the coil with enough power for it to work when the card is within a couple of inches from the coil; I cannot see it working effectively with a very large coil - the coupling to the card (which contains a resonant coil circuit) would be very weak and the energy returned from the card weaker still, to the point of not being detectable with such a simple circuit.
The coil in a normal card reader and coil in the card act as a transformer when the two similar size coils are close together; the close spacing also means it's natural to present the card flat-on to the reader
A card could have any orientation relative to a massive 1.5m coiI. I would not say that long distance reading is impossible, but I'm about 99.999% sure that simple fixed-threshold reader would not work.
For reference the formula for resonant frequency of a capacitor and inductor tuned circuit is:
1 / (2 x pi x square root (L x C))
L is in henries, C is in farads.
The formulas for working out the inductance of a coil from physical dimensions are rather more complex and I can never remember those...
When reading a card, the "transformer effect" between the two coils, the reader and the card, passes power to activate the electronics in the card.
To send the card serial number back to the reader, the electronics in the card switch a load (eg. a resistor) in and out of circuit to change the amount of power drawn through the transformer on a cycle-by-cycle basis and that affects the voltage on the transformer coil in the reader - as in picture 4 & 5 in your link.
Edit - the wikipedia article here has pictures of a proximity card that's been cracked apart.
You can see the coil in it that makes up the other half of the "transformer" when that is close to the correct reader coil:
Edit 2:
If you just want to make a reader to experiment with, you would probably be better off with one of these:
They are a ready built module which has a HTRC110 rfid reader IC plus a microcontroller to manage it.
All you need is the antenna coil, power and something to read the serial data the module sends out.
Data on the reader IC that uses, for reference - but you do not need to know anything about how this works to use the complete module. It's likely far more sensitive than the design you have.