Actually, that's not technically true. FWIU, Lighting could be run on both Bitcoin and Litecoin today, but issues with transaction malleability could cause it to be less than trustworthy, for some use cases. So, Segwit, which fixes transaction mailability by moving the header in the transaction to it's own section, enables a higher trust Lightning network to be deployed, on top of a more trustworthy blockchain.
Your understanding is correct AFAIK. However, the entire point of Bitcoin (and cryptocurrencies in general) is that it's a system that does not have to rely on trust. A "less than trustworthy" implementation is not a tenable implementation. In order for L2 solutions to be truly, universally usable we need a fix for transaction malleability.