The logo that appears at 0:39 is custom-drawn, though it’s obviously related to the logo used in much of the original marketing. I wonder if it was a candidate at one point for the logo to be used with the title crawl (which had white text in early test versions).
Also, those white lightsaber sticks are what the original in-camera lightsaber effect (using glowing sticks powered by electric cords hidden up the actors’ sleeves) looked like. It just didn’t work very well.
I knew about the lightsaber effects, but I just assumed that the logo was taken straight from one of the early designs. Though I haven’t seen this exact version, it seem like they just traced over the Dan Perri logo used for the Tom Jung poster. Makes sense though that they had to make a “new” version for the trailer in order to do the masking effect. The white logo is probably the exact same one as seen later in the trailer but with a simple white background added instead of ANH footage.