Yes, he was the Lord of Rivendell
The face of Elrond was ageless, neither old nor young, though in it
was written the memory of many things both glad and sorrowful. His
hair was dark as the shadows of twilight, and upon it was set a
circlet of silver; his eyes were grey as a clear evening, and in them
was a light like the light of stars. Venerable he seemed as a king
crowned with many winters, and yet hale as a tried warrior in the
fullness of his strength. He was the Lord of Rivendell and mighty
among both Elves and Men. - FOTR : Chapter 1 - Many Meetings
We are advised in the Unfinished Tales that he was appointed as Gil-galad's vice-regent in Eriador.
At this time the first Council was held, and it was there determined
that an Elvish stronghold in the east of Eriador should be maintained
at Imladris rather than in Eregion. At that time also Gil-galad gave
Vilya, the Blue Ring, to Elrond, and appointed him to be his
vice-regent in Eriador; but the Red Ring he kept, until he gave it to
Círdan when he set out from Lindon in the days of the Last Alliance.
And he also served as the herald of Gil-galad during the War of the Ring.
`I was the herald of Gil-galad and marched with his host. I was at the Battle of Dagorlad before the Black Gate of Mordor, where we had the mastery: for the Spear of Gil-galad and the Sword of Elendil, Aiglos and Narsil, none could withstand. FOTR : Chapter 2 - The Council of Elrond
The Silmarillion has some other (informal) titles for him; "master of Imladris" (note the lack of capitalisation) and "keeper of Vilya, the Ring of Air" as well as Master Elrond and Elrond Half-elven which is, as you've noted more of a descriptor:
Elrond Son of Eärendil and Elwing, who at the end of the First Age
chose to belong to the Firstborn, and remained in Middle-earth until
the end of the Third Age; master of Imladris (Rivendell) and keeper of
Vilya, the Ring of Air, which he had received from Gil-galad. Called
Master Elrond and Elrond Half-elven
Frodo, in Two Towers describes him as "Elrond of Imladris"
My part in the Company was known to him, as to all the others. for it
was appointed to me by Elrond of Imladris himself before the whole
Council. On that errand I came into this country, but it is not mine
to reveal to any outside the Company.
For the record, he is repeatedly called "Master Elrond" by the various members of the Fellowship including Frodo, Boromir and Gandalf:
'And let us hope that none will ever speak it here again,' answered
Gandalf. 'Nonetheless I do not ask your pardon, Master Elrond.