AMs have voted to rename the the assembly, calling it both Senedd Cymru and Welsh Parliament.
A majority backed former first minister Carwyn Jones’ proposals for a bilingual name, and rejected plans for a Welsh-only title, Senedd.
Plaid Cymru’s Rhun ap Iorwerth said the word Senedd transcends language barriers, but Mr Jones said it was not clear everyone understood it.
Presiding Officer Elin Jones said she was “disappointed” with the result.
A total of 43 AMs backed Mr Jones’ changes to the Senedd and Elections Bill, with 13 against.
A vote backing a Welsh-only name was defeated 16 votes for, 38 against, and one abstention.
It is not the final stage in the passage of Elin Jones’ bill through the assembly, but Wednesday’s votes were a key part of the process.
AMs also reiterated their support for votes for 16 and 17-year-olds in assembly elections.
A Brexit Party-bid backed by some Conservative AMs to scrap the plans failed to pass, with 11 votes for and 45 against.
After the bill passes the new name will come into force in May 2020, with changes to the voting age implemented at the next assembly elections in 2021.
The debate opened on Wednesday by Mr ap Iorwerth, who was supported by Labour’s Hefin David and Mike Hedges in tabling an amendment backing a Welsh-only name.
The Plaid AM said: “Some words transcend language barriers. There are some words in particular that in practical and usage terms genuinely belong to us all.
“Senedd is one of those words. A word that is of Wales; a word rooted in the Welsh language; a word that is bilingual in its application.
“What better application for such an inclusive word than as the official name of our national democratic and representative institution.”
Sian Gwenllian, of Plaid, highlighting Welsh words already used widely in Welsh English, said: “Let’s cwtch up today, let’s not be twp, let’s say together there’s a parliament for Wales to be known as Senedd.”
Bridgend AM Carwyn Jones’ proposals were supported by Welsh Government ministers.
He said he himself would use the term Senedd, but his amendments made it clear in law that Senedd Cymru was Welsh Parliament.
Carwyn Jones said it was “true to say Senedd is becoming more apparent among the public”.
“But I don’t think we are there yet, that everyone understands that Senedd means parliament”, he said.
“Maybe in years to come Senedd will be very common indeed,” the Labour politician added.
“But in the mean time I think it is important that people understand that this is the Senedd, but it is also the Welsh Parliament.”
The original bill stated that Senedd would be the name but that it could also be known as Welsh Parliament – ministers had been worried this would be confusing.
The former first minister was backed by David Melding, Conservative AM for South Wales Central.
He said a bilingual name would celebrate “both the magnificent worlds we live in, in the English speaking world and Welsh speaking world – that combination makes Wales a most exceptional place”.
Elin Jones’ law had already called the assembly Senedd, but had included a clause saying it “may” be known as the Welsh Parliament.
Mr ap Iorwerth’s amendment, which did not pass, tried to clarify that clause to make it clearer the name of the legislature would be Senedd.
Ms Jones said: “It is disappointing that the pre-eminence of the name Senedd has been defeated by a majority of assembly members.
“The term Senedd is both Welsh and international at the same time, and its simplicity has already secured its place in every day use.”
Osian Rhys from Cymdeithas yr Iaith, said it was “clear from the comments in the debate today that there is a lot of support across the parties for a Welsh-only name, ‘Senedd'”.
AMs to become Members of the Senedd?
Proposals to call AMs Members of Senedd Cymru or Member of Welsh Parliament were both rejected by assembly members – the former by a knife edge.
It means that, as things stand, AMs will be renamed Member of the Senedd as per the bill’s original proposal.
Mr Jones had proposed that members be called Members of Senedd Cymru, while Blaenau Gwent Labour AM Alun Davies had backed Members of Welsh Parliament.
Both were rejected. Mr Jones’ amendment for the new title failed to pass 25 votes for, 26 against, with four abstentions.
The votes meant that Mr Jones’ other successful changes left inconsistencies in the bill which will need to be changed at the next stage.
First Minister Mark Drakeford said it was a “marathon session” by the assembly with a “great deal” of AMs in debate.
He admitted there was a “little bit more work to do in refining the name of assembly members in future”.
Analysis by Felicity Evans, BBC Wales political editor
Watching this debate has been like going to a very intense game of bingo – trying to keeping track of more than 100 votes on hundreds of amendments to an already complex piece of legislation.
Reforming the way our national parliament works is not easy or simple.
Plenty of the amendments were about the new official title of the assembly and that appears to have been settled, but the new title for AMs remains up the air, I’m told.
The other matters were varied, everything from votes at 16 (which has had a lot of attention already) to giving voting rights to foreign nationals resident in Wales (which hasn’t).
That led to an accusation that Welsh Government was being “slipshod” and avoiding scrutiny by introducing the measure via an amendment.
Though it’s worth remembering that EU citizens are already eligible to vote in Welsh Assembly elections.
But this was only stage two of legislative progress. Even after a mammoth night, there’s still some way to go.
Foreign nationals to be given the right to vote in Wales
Later AMs voted for foreign nationals residing legally in Wales to be given the franchise in assembly elections – 38 for, 16 against.
Jeremy Miles, counsel general, told AMs: “Now is the right time for this institution to signal its commitment to people living in Wales regardless of where they were born.”
Mr Melding criticised the way the “major change” was introduced, “piggybacking” on an Assembly Commission bill. Mark Reckless said it was “impossible to conclude that it was for narrow partisan reasons”.
“And so is your opposition,” deputy culture minister Dafydd Elis Thomas heckled.
AMs also agreed measures to stop councillors from standing for the assembly, and to disqualify sex offenders from joining the institution.
Angela Burns, Tory AM, criticised the amount of time given over to debating the name, comparing it with 45 minutes in the chamber on Tuesday to discuss maternity services in Cwm Taf.
“I’m deeply uncomfortable by this imbalance,” she tweeted.