but LastPass is evidently designed with the assumption that each web site, network access point, etc., will have only one log-in dialog consisting of a user-name and a password. I don't have any criticism of the user-name and password assumption.

But there are numerous web sites for which there are two or more log-in dialogs. This is really common on merchant web sites, where one account is to "register" as a "shopper" and the other account is for transacting purchases. In most cases, it is up to the user whether they adopt the same user-name and password for each one, but usually they should have different passwords, at least.

For example, PC Tools has one log-in for the account which the customer creates to purchase the software. IIRC, it is also the source for downloading updates to it. But PC Tools does not have a tech-support staff. Instead, the customer needs to register at the "peer support forum" so that they can post messages asking for help, or post replies that offer advice to other users. The top-level domain name is exactly the same for the commerce account and for the support forum. LastPass cannot distinguish them and considers a record that was created for one to be a duplicate of the other, and deletes one of them.

Another example: my e-mail account has 8 sub-accounts (out of 10 allowed). To log-in to use the web mail interface, though, I must go to exactly the same web site page and enter the user-name for the account or sub-account, and the corresponding password. The e-mail client (Thunderbird) handles this all very well, but LastPass cannot, because the log-in dialog page for every e-mail account and sub-account has the same top-level domain name.

FWIW, I've been in communication with them about this problem in particular, and I suppose that they are probably working on it, whether they have adopted any of my suggestions for a solution.