Finally I remembered how to upload larger sized photos from Photobucket instead of the small thumbnail versions!!
Anyway this is an interesting subject almost like "which came first, the chicken or the egg"? It does look by this 1954 dated photo below (Mathewson Tool Company prototype) that the receiver with the scope mount is labeled a T44 (someone wrote in ink T44E4), and in the later 1960 dated photos from M1Army it was labeled T44E4, which is also correct. And to complicate this further, the T36 trials rifle in 1950 had the scope mount provision.
According to the Blake Stevens book "US Rifle M14 from John Garand to the M21", here are a couple of interesting excerpts from the book about the development of the T44/T44E4 rifles that competed with the T48 and Fal, so bear with me:
1. The contract to design and build the first, short-receiver T44 rifles was thus in place at the Mathewson Tool Company with John Garand acting as consultant. Twelve rifles were contracted for; the first (batch) was to be ready by June , 1954, and the second by September 1st. The remaining 10 were to be ready soon after. This model was christened the T44E4.
2. By August of 1955, Springfield was "on schedule" with the production of its lot of 500 test T44E4 rifles (based on the Mathewson prototype).
3. The Springfield records show they actually assembled 510 T44E4's, eight of which were held back for endurance trials. The rest were ready by September for the Ordnance Corps previously scheduled test of rifles made by American manufacturing methods. They were serially numbered from 1,001 to 1,510 (all of which had non-chromed barrels).
4. At the Pentagon, the T44E4 was preferred (over the T48 and Fal), due to its similarities to the M1 rifle.
5. Consideration of all of the above factors (extensive testing against the T48 and Fal) led to only one logical conclusion: that the Army should classify the T44E4 rifle as standard. A new number is obtained, which in this case turned out to be M14 for the light barrel version and M15 for the heavy barrel version.
6. William H. Brucker, the Secretary of the Army, made the announcement of the adoption of the M14/M15 family on May 1, 1957.
7. While the original Mathewson Tool Company design of the T44 / T44E4 rifle pioneered a threaded and keyed attachment point for telescopic sights and night vision equipment, Springfield was (later) ordered by Ordnance Weapons Command at Rock Island to build only the "absolute essentials" (for trials) into the final production drawings (which omitted the attachment point for telescopic sights).
8. Prior to final mass production contracts by Springfield, TRW, H&R, and Winchester the attachment point for telescopic sights was incorporated from the Mathewson Tool Company design.
So, in summary, it appears from my reading of Blake Stevens book that the Mathewson Tool Company T44 prototype was quickly renamed the T44E4, it originally was designed for a scope attachment, and the non-scoped provision version trials rifle by Springfield was also called the T44E4. The receiver made by LRB appears to be a faithful reproduction (some cosmetic differences noted) of the Mathewson Tool Company T44 / T44E4 until it was renamed the M14.