13" laptop: HD or 4k?

I agree. Don't even think about 4K on a 13 inch screen. That's way past what the eye can resolve in any case, so it's just wasted (even if you have perfect 20/20 vision). Looks good on paper, zero added value in reality - just a lot of unnecessary extra problems.

The minimum screen size where 4K begins to be useful is 24 inch or so. It still won't matter for images, but text will benefit. The retina MacBooks aren't full 4K either, but 2880 x 1800 on a 15 inch screen.

And Per mentions a very good reason for avoiding these high-density screens altogether: it gets very difficult to determine sharpness at 1:1. That's important!

When I bought my current desktop monitor, I debated with myself for a long time whether to get a 24 inch 4K model at 3840 x 2160, or the corresponding 27 inch model at 2560 x 1440. Price wasn't a factor. In the end I got the 27 inch, and I have not regretted that for a second.

The problem with high resolution screens (4k and higher) is that everything looks sharp, even when it may not be sharp in reality. On a 27" screen, this might not be a problem, but on a 13" screen the pixels will be really tiny, and it may be impossible to establish if an image is really sharp, even when viewing at 1:1 (100%). This is the only magnification that will give you true representation of the image – one image pixel will be represented by one screen pixel. But on a high resolution screen, it may be misleading.

Also, consider that Lightroom is very resource hungry, and may be slow with a laptop. With a 4k screen you will need to work with the GPU enabled, and if you for some reason have to disable the GPU (this happens to a lot of people), it may render Lightroom useless.

So my advice is to get the 1920 x 1080 screen.

One difference 4K will make is that it requires an graphics card that is compatible with Lightroom, or you will see LIghtroom operate slowly. If you usually perform a lot of local adjustments (spot healing or brushing) then the 4K screen will be even slower, with or without the graphics card..

Do yourself a favor and get the HD screen.