Whenever I often take photos of illuminated lamps with my phone, I notice that they rarely capture the same color that our eyes can see them. I wonder if there are any digital cameras that are able to capture the true color of illuminated light sources without having to rely on photo editing?

At least for white light, I think any camera will do as long as the white balance is locked at a neutral color (I find that 4100-4200K works best usually). The problem is that cameras always try to “correct” the white balance, which causes the color of white light to become distorted. On a phone, there are apps that let you lock the white balance. Any digital camera worth its salt should have a way to do this as well.

The problem is, in theory you can capture any color accurately (if you usa a sensor that has exactly the same spectral response of its channels like the eye), but you can never display all colors accurately, it is mathemathically impossible without an ability to really accurately form any arbitrary spectrum.
In real life common standadrs rely on 3 channels (R, G, B; or their recalculated equivalents like YUV or so), so not enough information to form all colors correctly. So the system needs some compromise. It is optimized so at least most natural and common colors are displayed correctly (skin tones, nature,...).
The drawback is for certain spectrums the color distortion becomes quite significant, an example being the typical mercury radiation spectrum.
So you can in theory tune the system to display the mercury radiation correctly, but then it will suck for other colors, like e.g. skin color tones (notoriously hard to display, as people are especially sensitive on any error there, yet normally of way higher priority for correct display than back alley mercury lamp).

Plus if you take into account what was the historical driving force for picture capturing (the movie industry), the scary looking greenish hue quite matches the mood of the scenes where this lighting uses to prevail, so such color distortion becomes more a feature than a bug...