Your node (or the node you choose to connect to) defines what is bitcoin, according to you.
Everyone gets a vote on what bitcoin is, but the definition of bitcoin that you vote for is that defined by the code running on the node you choose to connect to the network through.
Running your own node let's you speak for yourself. If you connect to someone else's node you're effectively saying "I vote for whatever they're voting for".
The two parties calling toilet roll "bitcoin", are not actually using the bitcoin network. They're using the "toilet roll" network and simply calling it bitcoin.
Their transaction would not appear on my node, or the vast majority of any other bitcoin nodes and therefore it's not bitcoin by consensus.
If you don't run your own node, you're trusting that the node you're connecting to isn't actually running the "toilet roll" network and you didn't just sell your car for a 6 pack of Andrex. With Bitcoin you never have to trust - you are always able to (and encouraged to) verify, and this is what running your own node allows: verification, with no trust required.