Networking :: Setup Network Printer To Combination Machine

I have two Linux machines both running Debian (I do not want Windows to be involved at all). One is a desktop and one is a laptop. I desire to connect a not-yet-purchased printer/scanner combination machine only to the internal LAN via my router (Linksys WRT54G) via Ethernet cable, or via wireless if the printer/scanner has that capability. I want to be able to print to that printer (and scan from its scanner) by only having to turn on the printer/scanner and only one of the other computers, and not have to have both Linux machines turned on in order to print and/or scan.

So my questions are:
When I look at specific models of printers, what should I be looking for in their specifications that indicates that this configuration is possible (i.e., should I be looking exclusively for printers that say they are "wireless printers" or "network printers")?The reason I ask this question: In my online searching, I thought "networked printer" or "stand-alone network printer" meant just that, but what I found instead are pages and pages of instructions on how to connect the printer locally to a Linux machine, with the associated setup to allow that Linux machine to serve print requests coming from other machines on the LAN. And that approach is not what I want to do (with the notable exception of temporary setup to validate that the printer is responding to requests for printing and scanning from Linux).

Are there particular brands, makes, or models that I would have better luck in getting to work in this manner (i.e., "better stick with HP or Xerox")? Am I kidding myself that a combination printer/scanner would work in the same fashion and still be relatively easy to set up on both Linux machines without resorting to some Windows-centric approach? Result of my searching so far (I have not worked my way through all of these in detail, but plan to): References to local printer connection which is not what I want:
Setting_Up_a_Network_Printer_using_CUPS
Set up a network printer using cups
Set up a printer

The post inside Setting_Up_a_Network_Printer_using_CUPS that starts with "running an HP Photosmart 8450 as a stand-alone networked printer" (where is the permalink?) is as close as I could get, but I am concerned that the instructions given are specific to the HP Photosmart 8450, or specific to the HP vendor, versus for all printers that can be connected to an Ethernet network (not that being locked into HP is going to be a problem necessarily, but I would like to know why if that is the case). Linux compatible printers says "Have a look at LinuxPrinting for known working drivers for printers data base. Also buy from a linux friendly company, ie HP, Brother, Epsom." Later in that thread, someone said Definitely don't buy canon.