Do the sentences marked by my red and blue arrows refer to 2 different manners of articulation? To wit, the paragraph in the screenshot presents 3 different manners of articulation?
If yes to 1, is there any picture for the blue articulation? Or can someone kindly draw one? I tried to phonate it, but my phonation doesn't sound like [ɹ].
All of those variants are variants within the same manner, which is "approximant" (central approximant, not lateral). They refer to more precise characterizations of "place" including secondary articulation. On p. 235 of The sounds of the world's languages there is a drawing of the articulation of [ɹ] for 6 American English speakers, and you can see that it is quite variable in articulation.
The passage you provided describes two (not three) realizations of the English phoneme /ɹ/. You may call them two "manners of articulation", but they aren't the manners of articulation as a technical term in phonetics. The manner of articulation, in the technical sense, of both realizations is approximant (may be fricative when devoiced, e.g. in "price").
The two variants technically differ in place of articulation, the one postalveolar and the other somewhere between palatal and velar (Laver 1994:302 calls it "pre-velar"), but since they sound virtually identical and an articulatory investigation such as MRI or ultrasound would be required to tell you which, the distinction is rarely made and they are both usually treated as postalveolar and transcribed as [ɹ]. Some authors treat the apical variant as a retroflex approximant [ɻ]. Also note that the lip rounding and the retraction of the tongue root which your source describes (none, either, or both of which may accompany either realization as far as I can tell) are technically secondary articulations (labialization and pharyngealization, repsectively), but they are often disregarded in literature for the sake of simplicity or because they are allophonic (i.e. optional).
In reality, there are various shades of articulation between them (from Tiede 2007):
Produce a uvular trill. Note that in order to do this you have to form a longitudinal furrow in the tongue within which the uvula vibrates. Now move the whole body of the tongue slightly forward, while retaining precisely that deeply furrowed configuration. The result should be a close approximation to the typical American ‘bird vowel’, for which the phonetic symbols [ɝ] and [ɚ] have been used—both representing a central vowel with an r-like modification.
Related question & answer: Apical postalveolar approximant [ɹ̺] and retroflex approximant [ɻ]: What is the difference?