Odds are good that your elms are American elm, from the big one right over-head. BUT, Siberian elm and all the other elm species have winged seeds that can blow long distances, there is a chance that the seedlings are not the same as the big tree near by. But odds are good that they are the same species.
You may just have to accept that you will never know for certain which elm species these seedlings are. Just call them elms until you figure it out. If you can find a key for elms, the traits to note are the pattern of hairs on the leaves, top (abaxial) and bottom (adaxial) surfaces. The pattern the protective bud scales make on the winter buds. Also the pattern of the vascular bundle scars in the abscission scars when the leaves fall off in autumn. These are the traits that without seed, will separate American elm from Slippery elm from Siberian elm from introduced Chinese elms and other elms. Botanical keys are a series of binary questions about traits. They lead to species determination. If you answer that a trait is present, the key will send you to the next numbered question about a trait. If you answer no, it will jump you to a different question about traits. Usually in less than 5 or 6 questions (traits present or not present) a species name can be determined. Rarely will the key run out to 10 or more questions. They are how "official botanists" determine what species they are looking at. Keys tend to be jargon heavy, requiring pausing to look words up. Difficult to use for laypersons. But they are definitive.
To make it more confusing, elms hybridize relatively easily.
You might just have to accept "Elm" or Ulmus species not determined.
The bark of more mature trees will help. American elm and Siberian elm have fairly coarse, rough bark that begins to develop around 10 years. Chinese elm typically has smooth bark that exfoliates to reveal patterns of different colors.
All my dogs came from the animal shelters, your elms are essentially "pound puppies", the pedigree might be pure, but with all stuff from the wild, you have to do serious taxonomy to be certain they are a specific species.
Almost every county in North American has more than one native species of elm and at least one or more species of invasive species elm from Europe of Asia. They are difficult to sort out.
Siberian elm, Ulmus pumila has the smallest leaves of any of the invasive Ulmus species in North America. There are a couple cultivars of Chinese elm that have smaller leaves, but these need to be rooted from cuttings, as they do not breed true from seed. American elm, can have pretty large leaves as can Ulmus rubra, the slippery elm.