it's actually worse than that - lightning only works in the context of centralized banking. It presumes there is some other entity with whom you can hold a channel open, who you do roughly equivalent amounts of deposits and withdrawals from. If that criterion is not met, then you have to keep closing channels and re-opening new ones, each of which incurs a new transaction fee just like if you did it yourself. So it is still expensive to send arbitrary transactions, it is just now relatively cheap to do "banking" with a single entity.
That entity sending money for you may still be expensive as well. If they have another entity who they do roughly equal amounts of transactions and withdrawals from then yes, they can nominally transact that for free, but they will still probably charge you for the service, and having that bank send a payment to an arbitrary user with whom they don't have balanced deposits/withdrawals incurs the normal bitcoin transaction fee. You can see where all of this is going - Lightning doesn't really solve anything, it still is only optimized for transactions between large, centralized entities.
Oh, and if you ever lose power or internet then your channel has to be closed and resolved, then reopened if you want to do further business. So you have to pay a transaction fee every time you lose internet (or there is a routing problem somewhere on the net) or power.
