AMs have backed renaming the assembly, calling it both Senedd Cymru and Welsh Parliament, in a vote on Wednesday.
A majority of AMs rejected a Welsh-only name, Senedd, and backed former first-minister Carwyn Jones’ proposals for a bilingual name.
Plaid’s Rhun ap Iorwerth said the word Senedd was rooted in the Welsh language, but Mr Jones said it was not clear everyone understood the word.
A total of 43 AMs backed Mr Jones’ plan, with 13 against.
Meanwhile AMs have reiterated their support for votes for 16 to 17-year-olds in assembly elections.
A Brexit Party attempt to scrap the plans from the Senedd and Elections Bill, understood to be backed by some Conservative AMs, failed – 11 votes for and 45 against.
It is not the final stage in the passage of the bill through the assembly, but is a major part of the process.
Former first minister Carwyn Jones’ proposal for a bilingual name had sparked debate – supporters of the name Senedd have included BBC News presenter Huw Edwards.
The debate opened on Wednesday with Mr ap Iorwerth, who was backed by Labour’s Hefin David and Mike Hedges in wanting to call the assembly simply Senedd.
The name is already used for the building in Cardiff Bay that houses the assembly chamber.
Mr ap Iorwerth said Senedd was a “word that is of Wales, a word rooted in the Welsh language, a word that is bilingual in its application”.
He said it “belongs to us all”.
Sian Gwenllian, of Plaid Cymru, 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.”
Mr Jones, Labour AM for Bridgend, had been supported by the Welsh Government.
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 argued that it was important that before the use of Senedd became common, that people understand the institution is the Welsh Parliament.
He said it was “true to say Senedd is becoming more apparent among the public”. But it was not the case yet that “everyone understands that Senedd means 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 “the magnificent world we live in, in the English speaking world and Welsh speaking world – that combination makes Wales an exceptional place”.
Labour’s Mike Hedges voted against the bi-lingual name preferring the institution to be called the Senedd. He told BBC Wales: “I’m disappointed that the name Senedd-only was rejected but I expect Senedd to be widely used in public discussion and by the media”.
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'”.
“Politicians are only half way through the law-making process, so there will be another chance to push an amendment to ensure one name for the Senedd in a few weeks.”
Proposals to call AMs Members of Senedd Cymru or Member of Welsh Parliament were both rejected – the former by a knife edge.
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.
It means that, as things stand, AMs will be renamed Member of the Senedd as per the bill’s original proposal.
The votes meant that Mr Jones’ other successful changes left inconsistencies in the bill which will need to be changed at the next stage.
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 a hundred 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 mammoth night, there’s still some way to go.
The Brexit Party made a failed bid to have the term Senedd removed from the English version of the bill, leaving simply Welsh Parliament.
David Rowlands, the party’s AM for South Wales Central, said: “It is essential that the name we use can be understood not only here in Wales but throughout the rest of the world.”
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 are continuing to vote on the bill, and are due to debate whether to stop councillors from standing for the assembly, and disqualify sex offenders from joining the institution.
Angela Burns, Tory AM, criticised the amount of time given over to debating the name, comparing it to 45 minutes in the chamber on Tuesday to discuss maternity services in Cwm Taf.
“I’m deeply uncomfortable by this imbalance,” she tweeted.