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 by 38 AMs, with 16 in support 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.
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 was opened by Mr ap Iorwerth who tabled an amendment, supported by Labour's Hefin David and Mike Hedges, backing a Welsh-only name.So good for those Labour MPs who supported Y Senedd only name
The Plaid AM for Ynys Mon said: "Some words transcend language barriers. There are some words in particular that in practical and usage terms genuinely belong to us all.It just occurred to me that we could argue that using Y Senedd or The Senedd could be regarded as bilingual just as Eisteddfod has entered the English dictionary and we use The Eisteddfod and not The Session .
"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."
Also using Welsh Senedd instead of Senedd Cymru would be bilingual names though i prefer Y Senedd alone.
As Sian Gwenllian, Plaid AM for Arfon, highlighted Welsh words already used widely in Welsh English
"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 would use the term Senedd himself, but his amendments made it clear in law that Senedd Cymru meant the Welsh Parliament.
Mr Jones said it was
A Pathetic argument and he probably knows it"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."
"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 former first minister was backed by David Melding, Conservative AM for South Wales Central.Utter bollocks and there are many bilingual countries in the World , but Cymru is dominated by the English language and Senedd would announce to te woeld that we dio have another language than English
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".
Senedd is already the name of the building that houses the assembly debating chamber, and is the Welsh word for parliament.
The original bill stated that Senedd would be the new name, but that the assembly "may also be known" as the Welsh Parliament. Ministers had concerns this would be confusing.
Mr ap Iorwerth's amendment, which did not pass, tried to make the law clearer in stating the name of the institution would be Senedd.
Elin Jones said
: "It is disappointing that the pre-eminence of the name Senedd has been defeated by a majority of assembly members.Osian Rhys, from Cymdeithas yr Iaith, said it was
"The term Senedd is both Welsh and international at the same time, and its simplicity has already secured its place in every day use."
"clear from the comments in the debate today that there is a lot of support across the parties for a Welsh-only name, 'Senedd'".Proposals to call AMs members of Senedd Cymru (MSCs) or members of the Welsh Parliament (MWPs) were both rejected by assembly members - the former by a close vote.
It means that, as things stand, AMs will be renamed members of the Senedd as per the bill's original proposal.
Mr Jones had proposed that members be called MSCs, while Blaenau Gwent Labour AM Alun Davies had backed MWPs.
MWP Mr Davies who is bilingual is surley aware that wags will aquaint it with the Welsh word Twp meaning stupid or dull,
Both were voted down. Mr Jones' amendment failed to pass with 25 votes for, 26 against, and 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 interest from AMs.
He admitted there was a "little bit more work to do in refining the name of assembly members in future".
We can off course easily predict that the English and much of the Welsh Media will only use the term Welsh Parliament , just as they refer to theÉireann as the Irish Parliament while actually the Parliament is Oireachtas is only one part of the legislature .
They also refer to The Leo Varadkar as the "Irish Prime Minister" and there are calls for us to refer to Boris Johnson as the "UK Taoiseach" and Westminster as the "UK Dáil ".
For me I will refer to our legislature as Y Senedd at least if we have two houses and may have to come up with a new name for both.