[chris]

iPods were launched way before iPlayer.

It was just very in-vogue to stick “i” in front of things in the early to mid 2000s.

[W. Knight]

Turns out iPlayer was tried internationally a decade ago (as an iPad app - the Apple product connections!), but failed:

The BBC has offered subscriptions to an international version of its iPlayer streaming service since 2011, but in the next month or so it will close its doors. In case you're wondering why you've never heard of the iPlayer Global service, that's because despite being offered in 16 countries (including Canada), the service never came to the US and stopped expanding back in 2013.

From www.engadget.com 

BBC shuts down iPlayer service for subscribers outside the UK | Engadget
The BBC has offered subscriptions to an international version of its iPlayer str…
  Source: www.engadget.com

It was shut down while the BBC Brit/Earth/First trio was set to replace BBC Entertainment/Knowledge, both announced by Tim Davie (!):
www.bbc.co.uk 

BBC Worldwide CEO, Tim Davie, shares future vision for BBC Worldwide - Media Cen…
BBC Worldwide CEO, Tim Davie, today unveiled a new vision to build the BBCȁ…
  Source: www.bbc.co.uk

Its replacement at the time? The 'new-and-improved' BBC.com. Funny how things had changed from having a 'do-all' website to a dedicated streaming service for VOD.

Additionally, BBC.com, the international version of the BBC’s online site, will be transformed over the next three years, supporting the BBC’s recently stated goal of doubling global reach from 250m to 500m per week by 2022. This transformation will see a greater focus on video content, bringing together all BBC commercial online offerings in one destination. The new BBC.com will include a long-form video player and will represent a single digital route to market for BBC.com’s partners and advertisers.

---

I think the situation in Canada is even more of an odd construct. It was rebranded as BBC First in 2021 and shows mainly BBC produced programmes, but is wholly owned by Blue Ant Media who licence the brand from the BBC.

Perhaps there are more Anglophiles in Canada with an appetite for Brit-stuff than in the USA.

Picking up from here, that's not the only BBC-branded channel there: BBC Kids apparently was a thing as well between BBC Worldwide and Alliance Atlantis (later Knowledge Network).
youtu.be 

To make things even weirder, they had a teenage block named BBCK!
youtu.be 

[Chad Blake]

Isn't BBCi still used for some of the BBC's web addresses (such as the one they store image assets on)? The logo is long-gone though.

[KatsKaravan]

iPods were launched way before iPlayer.

It was just very in-vogue to stick “i” in front of things in the early to mid 2000s.

Apple had a whole line of iProducts back then. As well as the iPod there was the iBook, the iMac and also the iTV. That last one had to be changed. A certain TV station threatened to sue...

[Brekkie]

I completely forgot about BBCi with its strange itv inspired logo for their online services and I’m sure you’re right this is a big part of why they went with the iPlayer name in the first place.

As a brand however it does leave a lot to be desired with the full BBC iPlayer name coming in 6 syllables hence it getting shortened to iPlayer in conversation which then omits the BBC brand. 

If they are going with BBC Player for the international brand then it’s best to stick with it a guess so it becomes recognisable but it does seem a bit generic and seems to describe the software rather than the content on it. You don’t have Netflix Player you just have Netflix for example. 

Maybe something like BBC View, BBC Vision or even BBC TV would work as better brands for the international market.

I think ultimately it will just be branded "BBC".   It would make more sense than "Channel 4", especially as most programming seems to be promoted as airing on "BBC iPlayer" now with no reference to the linear outing, so promoting as "on the BBC" would arguably be more inclusive.

[cable]

iPods were launched way before iPlayer.

It was just very in-vogue to stick “i” in front of things in the early to mid 2000s.

Remember the NetGem iPlayer - www.radioandtelly.co.uk 

iPlayer and iPlayer+ Freeview Boxes Explored | Radio & Telly UK
This page contains information on the NetGem iPlayer and BT iPlayer set-top boxe…
  Source: www.radioandtelly.co.uk

[W. Knight]

Turns out BBC UKTV was launched on South African satellite dishes last year, with a neon colour palette:

youtu.be 
youtu.be 

Complete with the Gill Sans blocks, as most other BBC Studios channels do...

[Roger Darthwell]

www.broadbandtvnews.com 

BBC Entertainment confirms closure
BBC Entertainment will close in all markets on 31 March 2024, Broadband TV News …
  Source: www.broadbandtvnews.com

it has just been annouced that BBC Entertainment, the successor channel to BBC Prime will be closed down in all markets on the 31st of March 2024

[W. Knight]

BBC Player, which spiralled a round of discussion about the missing 'i' here, now looks more the same to their UK sister with a "vibrant" (their words, not mine) splash of paint:
m.facebook.com 

[harshy]

www.broadbandtvnews.com 
it has just been annouced that BBC Entertainment, the successor channel to BBC Prime will be closed down in all markets on the 31st of March 2024

I don’t think it even changed look in all its years in existence.