The specific numbers in the question are from CCIR 601 (see Wikipedia article).

If you convert RGB -> grayscale with slightly different numbers / different methods, you won't see much difference at all on a normal computer screen under normal lighting conditions -- try it.

Here are some more links on color in general:

Wikipedia Luma

Bruce Lindbloom 's outstanding web site

chapter 4 on Color in the book by Colin Ware, "Information Visualization", isbn 1-55860-819-2; this long link to Ware in books.google.com may or may not work

cambridgeincolor : excellent, well-written "tutorials on how to acquire, interpret and process digital photographs using a visually-oriented approach that emphasizes concept over procedure"

Should you run into "linear" vs "nonlinear" RGB, here's part of an old note to myself on this. Repeat, in practice you won't see much difference.

In color science, the common RGB values, as in html rgb( 10%, 20%, 30% ), are called "nonlinear" or Gamma corrected. "Linear" values are defined as

Rlin = R^gamma,  Glin = G^gamma,  Blin = B^gamma

where gamma is 2.2 for many PCs. The usual R G B are sometimes written as R' G' B' (R' = Rlin ^ (1/gamma)) (purists tongue-click) but here I'll drop the '.

Brightness on a CRT display is proportional to RGBlin = RGB ^ gamma, so 50% gray on a CRT is quite dark: .5 ^ 2.2 = 22% of maximum brightness. (LCD displays are more complex; furthermore, some graphics cards compensate for gamma.)

To get the measure of lightness called L* from RGB, first divide R G B by 255, and compute

Y = .2126 * R^gamma + .7152 * G^gamma + .0722 * B^gamma

This is Y in XYZ color space; it is a measure of color "luminance". (The real formulas are not exactly x^gamma, but close; stick with x^gamma for a first pass.)

Finally,

L* = 116 * Y ^ 1/3 - 16

"... aspires to perceptual uniformity [and] closely matches human perception of lightness." -- Wikipedia Lab color space

How about doing it with Pillow:

from PIL import Image
img = Image.open('image.png').convert('L')
img.save('greyscale.png')

If an alpha (transparency) channel is present in the input image and should be preserved, use mode LA:

img = Image.open('image.png').convert('LA')

Using matplotlib and the formula

Y' = 0.2989 R + 0.5870 G + 0.1140 B 

you could do:

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.image as mpimg

def rgb2gray(rgb):
    return np.dot(rgb[...,:3], [0.2989, 0.5870, 0.1140])

img = mpimg.imread('image.png')     
gray = rgb2gray(img)    
plt.imshow(gray, cmap=plt.get_cmap('gray'), vmin=0, vmax=1)
plt.show()

The problem is that I can't understand how the "official" JPG->PGM convertitors work in terms of what value to assign to the final pixel (i guess, 0->255) starting from the classic RGB format.

There is likely a gamma adjustment in the conversion those "official" tools are using.
That is, it is not just a linear transform.

See this Wikipedia section for the details: Converting color to grayscale

I believe you want to use the formula for Csrgb.
Try it out and see if it matches the results you're expecting.

Basically, you'll do this:

1. Take R, G, B color (each in [0,1] range)
   • If they're in the range 0..255 instead, simply divide by 255.0
2. Compute Clinear = 0.2126 R + 0.7152 G + 0.0722 B
   • This is likely the linear transform you were applying before
3. Compute Csrgb according to it's formula, based on Clinear
   • This is the nonlinear gamma correction piece you were missing
   • Check out this WolframAlpha plot
   • Csrgb = 12.92 Clinear when Clinear <= 0.0031308
   • Csrgb = 1.055 Clinear1/2.4 - 0.055 when Clinear > 0.0031308