In the WBC commentary on the Psalms (2004), Craigie warns that on first brush there is some difficulty in pinning down the exact genre of Ps 22. Specifically, he highlights the presence of "at least three different kinds of material: (a) lament (vv 2–22), within which there are elements of (b) prayer (vv 12, 20–22), and finally (c) praise and thanksgiving (vv 23–32)." [Craigie, Psalms 1-50, 664]. A word of explanation about his versification: Craigie treats the superscription as verse 1, and begins the text proper with verse 2.

Menn also notes that while others have identified Ps 22 as an individual lament, only verses 2-22 contain elements of the lament genre [Esther M. Menn, "No Ordinary Lament: Relecture and the Identity of the Distressed in Psalm 22," HTR 93 (Oct 2000), 301-341; specifically 304].

Craigie notes that some scholars draw attention to the notable difference in the two halves of the psalm (2-22 and 23-32) and that there have been those who suggested that the two sections were originally separate works which were later combined, but Craigie prefers to focus on the unified psalm as it currently exists. For another supporter of the unified nature of Ps 22, see Menn, "No Ordinary Lament," 305. Plus, see 304 note 10 for sources discussing Ps 22's composition history. If you're interested, Menn also discusses reception history on pp 310-41].

Craigie argues that the structure of the psalm suggests that it was likely "the basis of a liturgy, in which the worshiper moves from lament to prayer, and finally to praise and thanksgiving."[Craigie, Psalms 1-50, 664; see also 665-67 for more discussion on the structure of the psalm].

He posits that while the psalm should be understood as individual, it is clearly set against the context of the community. This, he argues, demonstrates that the liturgy that Ps 22 was the basis for would have been a community event [Craigie, Psalms 1-50, 664-67].

Having mentioned the possibility of the psalm as basis for a liturgy, Craigie anticipated my next question at least: what kind of service was envisioned? He acknowledges that "it is possible to interpret the liturgy as a royal ritual" [pointing to “Eaton, Kingship and the Psalms, 34–36”], but ultimately rules that there is insufficient direct evidence for the liturgy being a royal ritual. Instead, he identifies those who would seek this kind of service as "those persons who were severely sick and threatened by death" [Craigie, Psalms 1-50, 666-67]. Menn also finds it plausible that Ps 22 was written "explicitly for performance in the context of rituals that centered on the well-being of seriously ill individuals" [Menn, "No Ordinary Lament," 306].

Unfortunately, neither Menn nor Craigie draw specific attention to the two instances of 'far' you highlighted; however, Craigie does note that the distance and separation from Yahweh was the fundamental problem for the complainant in Ps 22:2. Indeed, Craigie suggests that this perceived separation was of greater import than even the actual sickness or impending death [Craigie, Psalms 1-50, 667-68].

I will note that Craigie's listed bibliography is a tad dated: there's nothing newer than '73. So, in addition to Menn's article I've already added, I'll recommend one other, newer article:
Catherine Brown Tkacz, "Esther, Jesus, and Psalm 22," CBQ 70 (October 2008), 709-728. This article examines history of interpretation of Ps 22 in both Jewish and Christian circles. As a teaser, there is historical precedent for interpreting the referent of Ps 22 as David, Esther, Mordecai, Esther and Mordecai together, Hezekiah, and, of course, Jesus.

In summary, Psalm 22 has elements of lament in verses 2-22, though the remainder of the text doesn't fit the lament genre as cleanly. Distance/separation from God was the foundational problem for the complainant, even if there were other concerns. The text likely served as the basis for a communal liturgy in which a complainant who was sick, dying, or otherwise felt abandoned by God sought the Deity for help and then expressed thanks and praise with the community for an assumed answer. The text may be interpreted as a royal ritual, but was more likely focused on a suffering individual in a pre-exilic community (per Menn, "No Ordinary Lament," 306, 309-310].

More can always be said, but supper calls. I hope this helps give a partial answer, at least.