I think that I can identify your "No-Name" drill as a Black & Decker, by comparison to a B&D "Standard" 3/8" drill that I have. Despite the difference in size, mine has the steel rear cover plates, and the divided "D" handle with swinging trigger and pushbutton lock. I will post some photos to show why I think that mine is the "little brother" to yours.
The tape over part of the plates was there when I bought it, as was the price tag. The green Post-Tt (tm) tape carries my sequential number; in other words, this is the 238th electric drill that I have studied in enough detail to be able to get brand, manufacturer, date (if known), general features, power requirements, speed etc., etc., etc. for my collection's documentation.