Technically speaking, yes, there is a difference. Online refers to a remote connection to most any computer network or service whereas on the internet refers specifically to the publicly accessible network that got its start as a US defense department project known as ARPANET (along with other projects, to be fair).
In the past, there were services such as CompuServe, America OnLine (AOL), and FidoNet as well as private services -- in the late '80s, I was the system manager of a company where timesharing was a significant part of their income. (These days, the same concept -- storing your data and using programs on someone else's computer via a remote connection is called the cloud.)
So, yes, back in the day, you could go online to CompuServe or AOL or a Fido (or other) BBS without being connected to the internet. Even today, there are systems and networks you can connect to via a phone line or other means without being connected to the internet. (Note that internal networks, whether or not connected to the larger, external internet are called intranets.)
Note also that online has other meanings. The Oxford Living Dictionaries includes this adverbial definition:
In or into operation or existence.
‘the new power plant will go online this month’
‘the company has additional production capacity coming online later this year’
That said, if you were speaking to anyone other than an old, pedantic, computer history nerd like me, the assumption would be that online means the same thing as on the internet (primarily because most people have no idea that there is any other way to go online.)
In your specific situation, either option would be fine but, contrary to what your friend said, your use of on the internet is actually less ambiguous than their suggestion of online.