Can't Access Router Settings via 192.168.2.1

I have recently changed modem/routers as my connection speed was increased recently and the old modem wasn't up to the task... The previous modem/router was preconfigured and I did nothing to it, however the new one requires settings be fixed by accessing it via a browser at 192.168.2.1
When the technician came to check my line and set up the new modem, we encountered a strange situation. Despite being connected by Ethernet cable, my machine could not access the IP address reserved for the router.
The reply is: "Safari can’t connect to the server.
Safari can’t open the page “http://192.168.2.1/” because Safari can’t connect to the server “192.168.2.1”."
We tried Firefox and the same happened.
I have deactivated my firewall, rebooted, changed cables, restarted the router, got Safari to pretend to be Explorer, got the neighbor's dog to bark at it, everything except make a blood pact with the devil, and the two browsers on my machine think that 192.168.2.1 is an external address- which it isn't, and of course fails to connect.
The technician used his laptop to configure the thing, so I can access the internet (obviously) but I must find out why I can't access the router/modem's settings, if only to find some logical reason for this illogical behavior.
I've tried looking this issue up, and I get lots of bad wiring, wireless connection, windows proprietary shenanigans, but none that explain my situation.
I'd appreciate it if someone could help me make sense of this.
My setup:
Computer Model Name: Power Mac G4 (AGP graphics)
Model Identifier: PowerMac3,1
Processor Name: PowerPC G4 (3.3)
Processor Speed: 1.35 GHz
Number Of CPUs: 1
L2 Cache (per CPU): 256 KB
L3 Cache (per CPU): 2 MB
Memory: 832 MB
Bus Speed: 100 MHz
Boot ROM Version: 4.2.8f1
Sales Order Number: M7827LL/A
Router: "2-Wire Gateway"
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Your advice did the job quite nicely and it seems this is the "natural way of things," but I was totally unaware of it since my previous modems were preconfigured and I never had to muck about with subnets and such things.
What irks me is that this is basic, bare bones information that the Tech support people should have given me right off the bat. Why do they assume I have all the right numbers input into the Ethernet or PPPoE settings? They seemed totally perplexed that I couldn't connect, and frankly so was I- since my old Mac, as decrepit as it is, is still able to kick it with the modern way of things better than most computers I work with.
If they had asked me if I had set the right IP's in the right slots, it could have been fixed in 2 minutes.
Thank you very much for solving my problem, it saves me a lot of stress and confusion.