Yes, I'm pretty sure LastPass does handle it, but I haven't tried it.
Path1 and Path2 do not land on the same login form. It's effectively a different "site" (on the same domain) in that you see a different login page (different name, different graphics, etc). They are two distinct "sites" under the same domain.
If you inadvertently enter the credentials for the "other" form, you are logged in to the "other" site, which is very tricky if you're not paying attention. This is why you always have to know where you are (which isn't difficult), remember which credentials are appropriate (which is), and then select the right credentials if it happens they're not the ones already showing.
It's definitely possible to bookmark each one, and I have, but that doesn't help much with remembering the right credentials set.