That link implies you do not need a licence if you are watching on the iPlayer and quotes from that site saying one is not needed. However from the FAQs on that site it states clearly,
Catch-up
You do not need a television licence to catch-up on television programmes in BBC iPlayer, only when you watch or record at the same time (or virtually the same time) as it is being broadcast or otherwise distributed to the public. In BBC iPlayer, this is through the Watch Live simulcast option.
Live online (simulcasting)
Anyone in the UK watching or recording television as it's being broadcast or simulcast on any device - including mobiles, laptops and PCs - must, by law, be covered by a valid TV licence.
A 'live' TV programme is a programme, which is watched or recorded at the same time (or virtually the same time) as it is being broadcast or otherwise distributed to members of the public. As a general rule, if a person is watching a programme on a computer or other device at the same time as it is being shown on TV then the programme is 'live'. This is sometimes known as simulcasting.
The majority of UK households will already be covered by an existing television licence for their main TV set.