Every modern display supports DDC and related standards where it can tell the attached computer what its capabilities are.
In the absence of explicit configuration (e.g. by you, via the control panel), most operating systems will generate a signal corresponding to the display’s preferred resolution and refresh rate (typically matching the panel’s native capabilities), as reported by DDC.
According to the manual, a Dell U2713HM has a native resolution of 2560x1440. So that’s what macOS is going to default to until you configure it for something else. I assume the 1600x900 resolution you selected was manually configured.
OK. So the kind of cable is not an issue. Any HDMI cable should be able to support that resolution.
But…
If macOS thinks this is a “HiDPI”/Retina display, then it will be putting out a 2560x1440 signal, with the image scaled to be be equivalent to 1600x900. If that’s happening, then it will still need a high speed cable.
The cables that come with monitors aren’t always the best quality, but if it’s been working fine in the past, then it’s not going to be a manufacturing defect.
The fact that it is an 11 year old cable may mean that it is no longer up to spec. Conductors and contacts can oxidize over time. So you might want to try swapping its cable.
Another possibility: You’ve got two HDMI monitors. You said you are using adapters (docking station?). Maybe there’s an issue there?
The manual says that its inputs are DVI and VGA.
DVI, at the display’s native resolution, will require a dual-link interface, which a basic DVI-HDMI adapter won’t support. A lower resolution (like 1600x900) will have no problem with single-link, but not if HiDPI scaling is used. So double-check the settings and see if you can disable HiDPI operation.
For instance, my display shows this in the system settings (macOS 15.7.1):
Note that some resolutions (e.g. 1024x640) have both normal and HiDPI variants. The normal ones will produce a signal with that resolution. The HiDPI ones will produce the panel’s native resolution (1920x1200), with the image scaled (by the computer’s GPU) to be an equivalent size to the specified resolution.
If you’re using a HiDPI resolution, try switching it to the normal version of the same resolution.