I have worked around my issue with the display in a fashion that I didn't think possible with my lack of knowledge. So, the following is for people who experience the same problem or are just interested in the matter (conclusion at the bottom):
After searching for many, many different ways of describing my problem on Google, I came across this page on the Arduino forums of someone who had a completely different issue. However, Google found some text embedded in some code posted on that particular page (1.8" 128x160 SPI TFT LCD Display white screen - Displays - Arduino Forum), which had nothing to do with that problem, but was helpful for me:
// Our supplier changed the 1.8" display slightly after Jan 10, 2012
// so that the alignment of the TFT had to be shifted by a few pixels
// this just means the init code is slightly different. Check the
// color of the tab to see which init code to try. If the display is
// cut off or has extra 'random' pixels on the top & left, try the
// other option!
Now I don't have this particular display, but the description of the problem showed similarities to mine. And there was some sort of solution there as well. However, being the n00b I am, I understood next to nothing. I did give me the insight though, that I should try to make a workaround within the libraries that I will use in my programs. This way, I don't have to add extra code within the programs to shift the dimensions, and I can also download other programs and run them just fine with my altered libraries.
To make sure my display wasn't actually defect, I first looked for the option to broaden the resolution specifications, so I could see the pixels work. Instead of the usual 160x128 resolution, I compensated for the deviation with a resolution of 161x130: now all the pixels lit up as they should: no defect.
However, this solution would mean constantly accounting for a weird resolution which would make developing programs much more difficult than needed, since I would have to constantly remind myself of that odd resolution. Plus, there would always be extra columns and rows that recieved some computing, which would limit the speed of the Arduino. So I looked further in the libraries to find the place where the (0,0)-coordinates were defined.
The problem wasn't actually a problem within the files, so I suspect that there is indeed an alignment issue with my display. But I found the code within it, which I changed so that the starting point of the drawing shifted. After looking through all the libraries within the TFT folder (meaning: the TFT library, AdafruitGFX library and Adafruit ST7735 library) and trying to understand as much as I could, I found the location: within the Adafruit_ST7735.cpp file, there is code of the 'Adafruit_ST7735::commonInit(...)' function. This function defines the value of 'colstart' and 'rowstart' as 0. I changed it to correspond with my deviation.
I then of course set the resolution to default, because I don't want to have extra rows and columns on the other side of the display either.
Everything now works as it should have done in the first place.
After multiple hours of work, the description of the problem in the quote above seems very logical to me as how to solve it, but I like how I kind of figured it out by myself as well, knowing very little of programming languages.
1: Changing the resolution
A TFT display resolution can be configured within Adafruit_ST7735.cpp within the Gcmd array within the Adafruit_ST7735::writecommand(...) function. The other arrays in that function can also be configured, but TFT.cpp specifically states that a TFT display is configured according to the Gcmd array. I don't remember if it is necessary, or if I just added the following because I changed, tried and errored so much, but I also added the corresponding values to the '_width' and '_height' within the TFT.cpp file.
2: Changing the origin
The origin of a TFT display can be configured within Adafruit_ST7735.cpp within the 'Adafruit_ST7735::commonInit(...)' function. Changing the values of 'colstart' and 'rowstart' will change the row and column of the origin. By standard, they are both defined as 0 (-> colstart = rowstart = 0;), but writing them as two different definitions makes it possible to set a virtual origin, relative to the misaligned origin of the display.