In what sense does the security model of proof-of-stake require users to "log on to the internet every few months" This is primarily a question about something I heard Vitalik Buterin say in an interview the other day. (Unchained podcast's January 9, 2018 episode, at 34 minutes, 15 seconds) I quote:


  As far as the cons [of proof-of-stake], I would say the main one is that it is a different security model. It’s a different security model that basically says that, in order to securely authenticate the blockchain, you have to log on to the internet at least once every few months. And if you don’t do that, then in order to bootstrap yourself to the chain again you basically have to trust some group of people who have. And I think that in reality this is a totally fine security model because people basically already trust this security model for software updates.


I don't understand this quote. The way I understand it is that, regardless of proof-of-work (PoW) or proof-of-stake (PoS), a person has two choices: Run a full node, or run a "lite" node where you trust a third party node. 

If you run a full node, you have to download all of the chain before you can create new transactions. If you run a lite node then your balance and transactions are dependent on the third party, but you don't have to wait to download the chain. It seems to be that way both for clients of both PoW and PoS blockchains. So I don't see how the consensus algorithm is relevant. The only explanation is that I'm completely misunderstanding the quote, so I'm hoping that someone can shed some light on my misunderstanding.

Note: This question is not asking if PoW is better or worse than PoS. I just want to understand the quote. Thank you.
What Vitalik is talking about is the Slasher algorithm that he designed which punishes the block signer if he attempts to create a fork in the blockchain. However the way that slasher works is that there will be basically a timeout; if a fork is detected during that timeout, the signer that created the fork loses his money. But after the timeout, he does not. This is so that he can actually spend the money at some point.

The problem with this is that once the timeout has passed, the signer can then create a fork without being punished. This is known as a long range fork. This fork, if it survives, can grow and, at some point, could be longer than what you have synced up to. In that case, because of how PoS works, you could end up switching to using the forked chain instead of the real chain. So to avoid that, you need to go online routinely to keep your blockchain up to date so that in the event of a long range fork, you are still using the correct blockchain.