Did Jesus Die? (Psalm 116:13)

According to some Muslims, Psalm 116:13 shows that Jesus did not die; however, this is pure eisegesis. The verse reads: "I will lift up the cup of salvation and call on the name of the Lord," (Psalm 116:13 ESV). The claim is that the "cup of salvation" is a reference to the cross, for Luke says: "saying, “Father, if you are willing, remove this cup from me. Nevertheless, not my will, but yours, be done," (Luke 22:42 ESV).

The main issue with this verse is who is actually speaking in the Psalm. "I love the Lord, because he has heard my voice and my pleas for mercy. Because he inclined his ear to me, therefore I will call on him as long as I live. The snares of death encompassed me; the pangs of Sheol laid hold on me; I suffered distress and anguish," (Psalm 116:1-3 ESV). This is not God speaking, saying he will save the messiah, it is a psalmist speaking. Unless the Muslim is to claim this psalmist was God, this interpretation seems nonsensical.
Rather, this seems to be some sort of celebration of God. As Barnes explains: "The "cup of salvation" means the cup by which his sense of the greatness of the salvation might be expressed - the cup of thanksgiving. ...The reference seems to be to a custom in festivals of drinking a cup of wine as a special expression of thanks or of obligation. The act would be more solemn, and the truth more deeply impressed on the mind, when accompanied by some religious rite - some ceremonial, as in the Lord's Supper, expressly designed to call the mercy of God to remembrance," (Barnes).

Poole also writes: "the cup of blessing, 1 Corinthians 10:16, which is in effect the same with the cup of salvation. This metaphor of a cup is used both of afflictions, as Psalm 11:6 75:8, and of comforts, as Psalm 23:5 Jeremiah 16:7," (Poole). Jesus hoped the Father would take his cup, to reiterate what Poole had said, it is to ask for God to take away his afflictions.

References:

A Commentary on the Holy Bible. United Kingdom, Banner of Truth Trust, 1969.

Barnes' Notes on the New Testament. United Kingdom, Kregel Publications, 1962.