Its not a lightning wallet its an account!
But hey I'll give you props with the consistency. It is the same thing as wallet of satoshi for example after all.
Discussion
Practically speaking even the "non-custodial mobile wallets" are not really wallets, as @npub1ey6qdmvzcgcsr883m9nspzz0mm037l26xtardzcskfsvc6gc7jssm9szvp of Breez highlights in https://medium.com/breez-technology/there-is-no-such-thing-as-a-lightning-wallet-dc9ac46d32
The 3 hardest things in our business are:
1. Naming
2. Off-by-one errors
I'll give you that. However, in my mind, wallet means its on my person. Which yeah yeah its not only the ability to sign is on my person, but the ability to sign being on my person and not accessible to anyone else that I have defined as a wallet in my mind and if the ability to sign is on someone else and especially if I have to ask them for permission to spend then that's an account.
But damn we do need better words huh...