The phrase “the Lord of the Rings” is ambiguous in the same way in English: it might, in theory, refer either to the One Ring, which rules the other rings, or to Sauron, who can use it to rule all the rings.
However, the text does not make use of this ambiguity. The phrase “the Lord of the Ring(s)” appears four times in the text, and three times it’s clear that Sauron is intended:
‘Yes, I knew of them. Indeed I spoke of them once to you; for the Black Riders are the Ringwraiths, the Nine Servants of the Lord of the Rings.’
J. R. R. Tolkien (1954). The Fellowship of the Ring, book II, chapter 1. London: Allen & Unwin.
‘But in any case,’ said Glorfindel, ‘to send the Ring to [Bombadil] would only postpone the day of evil. He is far away. We could not now take it back to him, unguessed, unmarked by any spy. And even if we could, soon or late the Lord of the Rings would learn of its hiding place and would bend all his power towards it.’ [II.2]
Here Bilbo’s hand ended and Frodo had written:
LORD OF THE RINGS AND THE
RETURN OF THE KING
J. R. R. Tolkien (1955). The Return of the King, book VI, chapter 9. London: Allen & Unwin.
In the fourth case Pippin uses the phrase as a joke, and Gandalf quickly corrects him:
‘Hurray!’ cried Pippin, springing up. ‘Here is our noble cousin! Make way for Frodo, Lord of the Ring!’
‘Hush!’ said Gandalf from the shadows at the back of the porch. ‘Evil things do not come into this valley; but all the same we should not name them. The Lord of the Ring is not Frodo, but the master of the Dark Tower of Mordor, whose power is again stretching out over the world!’ [II.1]
When the characters want to emphasize the role of the One Ring in controlling the other rings, they use the phrase “Ruling Ring” instead:
‘It has been said that dragon-fire could melt and consume the Rings of Power, but there is not now any dragon left on earth in which the old fire is hot enough; nor was there ever any dragon, not even Ancalagon the Black, who could have harmed the One Ring, the Ruling Ring, for that was made by Sauron himself.’ [I.2]
‘For in the days of Isildur the Ruling Ring passed out of all knowledge, and the Three were released from its dominion.’ [II.2]
‘[Saruman] came and laid his long hand on my arm. “And why not, Gandalf?” he whispered. “Why not? The Ruling Ring? If we could command that, then the Power would pass to us.”’ [II.2]
‘Alas, no,’ said Elrond. ‘We cannot use the Ruling Ring. That we now know too well. It belongs to Sauron and was made by him alone, and is altogether evil.’ [II.2]
‘But what then would happen, if the Ruling Ring were destroyed as you counsel?’ asked Glóin. [II.2]
‘The Ringwraiths are deadly enemies, but they are only shadows yet of the power and terror they would possess if the Ruling Ring was on their master’s hand again.’ [II.4]
If Tolkien had chosen the title with an eye to its ambiguity, I would have expected him to make use of the ambiguity somewhere in the text, but he does not: he maintains a consistent distinction between “the Lord of the Rings” and “the Ruling Ring” to convey the two senses. This suggests that the title was not chosen on purpose to be ambiguous.