The full Greek reads:
1. Περὶ δὲ τοῦ βαπτίσματος, οὕτω βαπτίσατε· ταῦτα πάντα πρειπόντες, βαπτίσατε εἰς τὸ ὄνομα τοῦ πατρὸς καὶ τοῦ υἱοῦ καὶ τοῦ ἁγίου πνεύματος ἐν ὕδατι ζῶντι. 2. ἐὰν δὲ μὴ ἔχῃς ὕδωρ ζῶν, εἰς ἄλλο ὕδωρ βάπτισον· εἰ δ’ οὐ δύνασαι ἐν ψυχρῷ, ἐν θερμῷ. 3. ἐὰν δὲ ἀμφότερα μὴ ἔχῃς, ἔκχεον εἰς τὴν κεφαλὴν τρὶς ὕδωρ εἰς ὄνομα πατρὸς καὶ υἱοῦ καὶ ἁγίου πνεύματος. 4. πρὸ δὲ τοῦ βαπτίσμος προνηστευσάτω ὁ βαπτίζων καὶ ὁ βαπτιζόμενος καὶ εἴ τινες ἄλλοι δύναται· κελεύεις δὲ νηστεῦσαι τὸν βαπτιζόμενον πρὸ μιᾶς ἢ δύο.
And concerning
baptism,
baptize this way: Having first said all these things,
baptize into the name of the
Father, and of the
Son, and of the
Holy Spirit, in living water. But if you have not living water,
baptize into other water; and if you can not in cold, in warm. But if you have not either, pour out water thrice upon the head into the name of Father and Son and
Holy Spirit. But before the
baptism let the baptizer fast, and the
baptized, and whatever others can; but you shall order the
baptized to fast one or two days before.
So in this instance living water means a moving body of water, aka a river or a spring. So it is pointing to following the examples of Jesus being baptized in the Jordan and the Ethiopian eunuch. So the Didache gave some alternatives since someone might not be near an actual river.
This is not about Jesus reference of "Living Water" but rather a practical document. If you aren't near a river, use something else. Worst case, pour water over the head three times. So for example, my church does not have a baptismal pool but a large font so an adult leans over and the priest pours a large pitcher of water over their head three times.
