Remove the jumpers, they're superfluous
It appears from the installation manual for your zone controller that the C terminals for the thermostats are connected internally to the C wire coming into the board from its 24VAC power source. This means that the white jumper wires between the zone C terminals and the blue C wire from the furnace can be removed altogether and your C wires hooked up in their place.
As to the 1K resistor? Someone thought this unit power-stole, I guess, and put that in there to provide a bypass load to prevent erratic operation. I'd remove it and keep an eye on the system -- if it doesn't malfunction after a week or two without it, I'd say whoever put it in was a member of a power-stealing cargo cult.
Someone wired your thermostat in rather...backwards
Normally, the R terminal on a thermostat goes to the 24VAC end of the transformer, the W terminal on a thermostat goes to the TH terminal on whatever the thermostat is controlling, and the TR terminal on whatever is getting controlled goes back to the other (0VAC) end of the transformer, which is also where the C wire connects.
This is analogous to having a light switch in the hot wire to a light, with the R wire being the hot feed from the panel, the W wire being the switched hot to the light, the C wire being the neutral, the thermostat taking the place of the switch, and the device under control assuming the role of the light.
However, in your case, the installer routed R to the 0VAC (C-wire) end of the transformer, and connected the TR terminal on the device under control to the 24VAC (R-wire) end of the transformer. This still works, as the switch and the device are in series, and neither is polarized because we're dealing with AC here -- its analogous to having a switch in the neutral wire. (The 120VAC version is against Code, despite the fact it works electrically, because it's a great way to get bit by 60Hz while changing a lightbulb.)
However, if your thermostat needs a C wire, things start to fall down a bit. Connecting C back to the 0VAC (C-wire) terminal on the transformer will make any of the fancy things on the thermostat that need the C-wire to work simply not work because there's no voltage between R and C in that case. Instead, in order to get the fancy stuff to work, you have to connect C on the thermostat to the 24VAC (R-wire) end of the transformer. This will still work, most likely, because it's an AC system and thus not polarized. However, it's going to be awkward for the next poor bloke who has to service this system!
It should be relatively simple to switch the wiring around at the junction where the wires for the master thermostat and it's associated zone valve are located. The R-wire from the 'stat should go to the 24VAC (R) end of the transformer like the other R-wires, and the wire from the TR terminal on the zone valve should go to the 0VAC (C) end of the transformer, just like how the other zone valves are wired up.
Best Answer
You just can't get what you want...
Unfortunately, each zone relay has its own independent 24VAC transformer, so trying to "poach" a C wire connection from another zone's relay simply won't work at all. No sense continuing down that dead end!
However, you just might be able to get what you need
The good news is that there is a way out of your situation, since the Ecobee model you have supports two-transformer systems. You'll need:
We first start by turning the HVAC system breaker off and mounting our 4" square box somewhere convenient near the existing zone relays, then running the 12/2 armored cable over to said 4" square box from the top of the leftmost zone relay using the "redhead" anti-short bushings and cableclamps. Once that's done, we can wirenut the transformer's 120V leads to the black and white wires on the armored cable, then mount the transformer. At the other end of the AC run, we crimp spade lugs onto the white and black wires then land them on the appropriate terminal screws on the zone relay board. Once those terminal screws are retightened, then we can move onto the secondary side of the transformer. The 18/2 cable attaches to the two terminal screws on the transformer at one end, and gets wirenutted to the green and blue wires on Zone 1's thermostat cable at the other end.
Now that we're done at the zone relays, we can tackle installing the new thermostat for zone 1. Green and blue go to Rc and C respectively, while red and white go to Rh and W/W1. This way, the Ecobee will draw power from the 24VAC transformer you just installed, while controlling the heat by connecting Rh to W to activate the zone relay by bridging the T-T terminals on said relay.