Hello, I'm posting this here with a hope that the bulb-lovers may have an answer to my query. I have these two in my yard. See the difference in flowers. I'm eager to know more about these.

I wonder if the 2nd one could be Peruvian daffodil Peruvian Daffodil (Ismene narcissiflora)
The first could be a species Spider Lily (Hymenocallis speciosa)

Here is my Peruvian Daffodil

It's not Ismene (they have an extra set of petals/sepals whatever that curl back heavily).

Most propably it's two different species, a variation in the genetic make up within the species, or - what I think it really is - age of the flower. Older flowers spread open much like the first photo and younger ones look more cup shaped like the second.

I've also noticed on some of my H. harrisiana having differences in flowers. Some had short pistils and others had much longer ones.
It all adds to the natural diversity.

In fact, I remember when I moved in here 19 years ago - my ancestral property - the tenant who was in this place for 47 years had this in a bunch. I separated them as they had crowded since many years perhaps. I had move them here and there but had not thought there were two different flowering types mixed up. I had a smaller lily which grew just a foot tall [leaves with a central ridge unlike the slender rain lily leaves] but bigger cup and short white petal. There were plenty of them but I had not known the importance of keeping them safe. I used to chop down the leaves as I did not want them - this was in my initial stages of gardening as an ignorant novice. Now I have saved two tiny bulbs and they are struggling with thin and weak leaves - short and with no energy whatsoever.
I'm sure this is not a daffodil. I wonder if a slight mutation has happened when the bulb was divided.

I took another look. The leaves of the cup-like flower are a little broader and have a lighter green colour whereas the other one has narrower, more rigid darker green leaves. My photo above also shows these.