The Hebrew that is translated as “Thus says the Lord, ‘By this you shall know that I am the Lord.’” or similar in English is translated in the Danish Bibelen på Hverdagsdansk (publ. 1985, rev. 2015 et al.) as “But now the God of the Hebrews [hebræernes Gud] is telling you: You will see that I am their God.”

Iver Larsen explains: “Each nation had their own god who should protect the people and fight their enemies. The people should then worship their god. Pharaoh knew his own gods, and YHWH was not one of them. However, he is told that this unknown god, called YHWH, is the God of the Hebrews, the God of Israel. YHWH will now show that he is fighting for his people, the Hebrews. He will also show that he is more powerful than the Egyptian gods. For the last sentence maybe we could say: By this you will know that I am the God of the Hebrews. The implication is that the Hebrews are his people and he is going to fight for them.”

Sabaot translate this phrase in connection with verse 16 as this: “When he arrives, tell him, ‘The God of the people of Israel has sent me to come and tell you to release my people to go and worship/bow to me there in the wilderness. But you have made your head hard (stubborn) until now. So, he has said that you shall today know/understand who/how he is through the amazing things he is going to do.” (Source: Iver Larsen)

See also says the LORD / says the Lord GOD.