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I googled some online tests that should test color vision problems. But then it got to me that any picture is rendered through the same 3 color pixels. Thus it seems - if one perceives the red pixel differently than the green pixel, that person will be able to tell apart the different combinations of those two.

Is it correct that the color-vision tests that are shown on computer screen will only be able to diagnose dichromacy (in the best case scenario) but not anomalous trichromacy?

Yes, is is possible to test for it, but with some very severe limitations for most classical tests, like Ishihara plates.

If you go to the Wikipedia page for Ishihara test you may see a demonstration of effects that will or will not apply to your personal type of vision. This gets all the more striking if you show those pictures (preferably all the plates of this test) to a person with known one type of known Daltonism.

But this is exactly limited by what your initial thoughts indicate.
Go for example to Color vision test, do the test with your standard screen, repeat the test with a known "bad screen". The test has a nice design but even people with some quite severe deficiencies can "beat it", all the more easier with bad screens, it they change the viewing angle (look from the side, tilt the screen).

The future of color vision testing
Today in our digital world one might think, why don’t we have some simple computer based color blindness test. Unfortunately this is not as simple as it looks like. There are two main problems:

1. Computers displays just make use of three main colors red, green and blue (RGB). Every other color gets mixed from those three colors. The anomaloscope and lantern tests use different light sources which can’t be simulated by a display.


2. Every computer display has a different color range it covers, little differences in light sources, different brightness and more. This causes different test results. Only calibrated computers can be used to perform such computer based tests.

The City University in London developed a computer based color vision test which is also based on the same principal as pseudoisochromatic plates and arrangement tests. The main difference is that the colors are constantly changing which gives some really good results. Just recently they used their test to check color vision in pilot candidates and it looks like as the Color Assessment & Diagnosis Test (CAD Test) could become a standard screening instrument for color vision testing. At least for certain professions, where color vision is critical but people with a mild form of color vision still perform perfectly.

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City University of London: A new web-based colour vision test

The web version of this test will run on a variety of monitors balanced for different phases of daylight. The movie was however prepared and will run best on a monitor balanced for ~9000K. This is usually the default factory setting for most colour monitors. The spectral characteristics of the pattern will be affected by ambient illumination and therefore this should be kept to a minimum (i.e., use the monitor in a dark room).

Note that even though the above test was developed specifically "for the web" only gives you a rough indication of "problems" with colour vision.

Aside from the problems mentioned in the quotes above, accurate colour is something very hard to buy hardware for. If they are even capable of displaying more than a compartively small subset of the full spectrum, most modern screens are tuned for 'pop' and not realistic colour reproduction:

If you walk into a Walmart, Best Buy, or any major TV retailer, all of the TVs are showing identical videos, but their colors will be noticeably (to obnoxiously) different on every TV on the wall – even in Best Buy’s high-end Magnolia showrooms. Why is that? One reason is that the TV picture controls have been played with – but this would still be true even with factory fresh TVs right out of the box. It’s also true with Smartphones and Tablets that as a rule don’t provide any color controls, which is probably better because visual tweaking generally makes matters worse. And that is exactly the root cause of the problem that originates right at the factory – the displays are not individually fully adjusted and calibrated with instruments, and instead depend on visually tweaking at some point during manufacturing.