Personal Visit There is no doubt, hermeneutically and exegetically, that God appeared to Abraham during the visit of the three men-looking individuals: the LORD and two angels. (Genesis 18)
However, it would be better, and most accurate, to label this event as a CHRISTOPHANY, than THEOPHANY. The appearance of God Almighty in His effulgent glory would be too overwhelming for any mortal to see and live! He is called "the God of glory" in Acts 7:2. Which is an understatement. Moses asked to "see God's glory" (Exodus 33:18), but he was "biting off more than he could chew." God allowed Moses to "see His goodness" in passing; but not "His face", that is His Person. Moses could only see "the departing of God."
This is reminiscent of God only letting one High Priest enter the Holy of Holies where the Shekinah Glory of God was over the Ark of the Covenant, between the cherubim...but only after thorough, ceremonial cleansing...with the blood of atonement. Anyone else would be slain!
Christophany But God appeared many times in the Old Testament...and eventually in the New Testament... in the form of Christ (also called The Angel of the Lord). This was a "bite-size portion of God Almighty that man could handle...that man could see and still live. Such that the Apostle John declared:
No one has ever seen God; but God the only begotten, who is at the Father's side, has made Him known. (John 1:18)
Jesus made the Father known through His appearances on earth. Christophanies in the Old Testament, and as the Son of Mary, in the New Testament. Indeed, Jesus appeared to Abraham...he saw God but didn't die. Jesus was a "form of God" that mortal men could handle safely. And Jesus was "God." Not just an angel...not just a mystical spirit...not a substitute image:
Philip said, "Lord, show us the Father and that will be enough for us." Jesus answered, "Don't you know Me...? Anyone who has seen Me has seen the Father...I am in the Father, and the Father is in Me... (John 14:8-10)
Jesus is the image of the invisible God, the firstborn over all creation. (Colossians 1:15)
The Son is the radiance of God's glory and the exact representation of His being sustaining all things by His powerful Word. (Hebrews 1:3)
The writer of Genesis can truly say that YHWH (the LORD) appeared to Abraham, even though it was in the form of a man: the Christ-man, who was...and is...fully God!
And this is also why Hagar can declare that she had "seen God" (Genesis 16:13). And it is also why Gideon could see the Angel of the Lord, and yet the LORD say that, even though he saw God, "he would not die." (Judges 6:23-24). It is also why the parents of Samson could "see God" and yet not die." "We are doomed to die," Manoah said to his wife. "We have seen God!" No, the "Angel of the LORD" (Christ) was in a form of God that mankind could handle. (Judges 13:22-23)
There is no doubt that Abraham experienced a Christophany: God in the flesh!