CSB 2 John 1:
9 Anyone who does not remain in Christ’s teaching but goes beyond it does not have God. The one who remains in that teaching, this one has both the Father and the Son.
10 If anyone comes to you and does not bring this teaching, do not receive him into your home, and don’t greet him;
[2Jo 1:9-10 MGNT] 9 πας ο προαγων και μη μενων εν τη διδαχη του χριστου θεον ουκ εχει ο μενων εν τη διδαχη ουτος και τον πατερα και τον υιον εχει 10 ει τις ερχεται προς υμας και ταυτην την διδαχην ου φερει μη λαμβανετε αυτον εις οικιαν και χαιρειν αυτω μη λεγετε
I believe that the latter ("the teachings that Jesus taught") rather than the former ("the teachings about Jesus") was intended. I think that if it were the former, the preposition περί would have been used, as in (with RSV translation):
And he lived there two whole years at his own expense, and welcomed all who came to him, preaching the kingdom of God and teaching about the Lord Jesus Christ quite openly and unhindered.
Ἔμεινε δὲ ὁ Παῦλος διετίαν ὅλην ἐν ἰδίῳ μισθώματι καὶ ἀπεδέχετο πάντας τοὺς εἰσπορευομένους πρὸς αὐτόν, κηρύσσων τὴν βασιλείαν τοῦ Θεοῦ καὶ διδάσκων τὰ περὶ τοῦ Κυρίου Ἰησοῦ Χριστοῦ μετὰ πάσης παρρησίας ἀκωλύτως.
1 John 2:27
But the anointing which you received from him abides in you, and you have no need that any one should teach you; as his anointing teaches you about everything, and is true, and is no lie, just as it has taught you, abide in him.
καὶ ὑμεῖς, τὸ χρῖσμα ὃ ἐλάβετε ἀπʼ αὐτοῦ, ἐν ὑμῖν μένει, καὶ οὐ χρείαν ἔχετε ἵνα τις διδάσκῃ ὑμᾶς, ἀλλʼ ὡς τὸ αὐτὸ χρῖσμα διδάσκει ὑμᾶς περὶ πάντων, καὶ ἀληθές ἐστι καὶ οὐκ ἔστι ψεῦδος, καὶ καθὼς ἐδίδαξεν ὑμᾶς μενεῖτε ἐν αὐτῷ.
The genitive case in Greek (as in English and many other languages) can mean any of the following:
- Belonging to someone, eg, This is the hand of David
- About or concerning someone, eg, This is the story of Moses
- Created by someone, eg, The cathedral of Gaudi
- Exclusive use (but not owned by) someone, eg, The chauffer-driven limousine of the president; or, the Palace of the Queen Elizabeth
- Conveyed by a messenger, eg, the message of the messenger about the king
An identical construction occurs frequently in the NT and one of the most discussed is Rev 1:1, "The revelation of Jesus" which is arguably a combination of (at least) the first three above.
In the case of 2 John 9, "the teaching of Jesus", we do not necessarily need (grammatically) to exclusively decide between these possibilities because, as illustrated above, it can, at times, legitimately mean more than one of the above. That is, in this case, we might have any of the following:
- The teaching of Jesus could be all the doctrines about Jesus' birth, death and resurrection and saving grace, etc; that is, the teaching ABOUT Jesus. Such teaching has been done by the apostles (2 Tim 1:13, 1 Tim 6:3)), and the Holy Spirit (John 16:13, 14).
- The teaching of Jesus could be that which was verbally taught by Jesus, eg, Matt 4:23, 7:28, 9:35, John 7:28, 18:19, etc. See also 1 John 2:27. That is, the teaching BY Jesus.
- The teaching of Jesus could also mean the teaching that Jesus conveyed about the God the Father (John 7:16). That is (also) the teaching BY Jesus.
It appears to me that 2 John 9 is making a distinction between he legitimate "teaching of Jesus" and that which is either extrapolated ("goes beyond") from it or is false teaching about Jesus. Such a statement is tantamount to what is "canonical" and would presumably include what we have recorded in the four canonical Gospels plus the material recorded for us by the apostles. Nothing else.
The NT contains several other such "canonical" claims such as the true Gospel (Gal 1:8) and revelation (Rev 22:18).
Therefore, I understand 2 John 9 to mean a combination of (1), (2), & (3) above and specifically restricted to these; that is, excluding anything that disagrees with these or extrapolates from them.