What does half-lit means? Aristarchus estimated the relative distance of the Sun and Moon by
  observing the angle between the Sun and the Moon (α in the diagram)
  when the Moon is exactly half lit. Angle β must be 90° for the Moon to
  be half lit. By observing the angle α, he could then set the scale of
  the triangle and thus the relative lengths of the sides. (Sizes and
  distances are not to scale.)


Does half-lit mean exactly 45 degrees?
Half lit means the Moon is at a quarter phase, either First Quarter or Third Quarter.

The original question asks "Does half-lit mean exactly 45 degrees?".  I assume this is asking if the angle $\alpha$ is 45 degrees. The answer is no. The angle $\alpha$ is found from trigonometry to be $cos(\alpha)=EM/ES$ where EM is the distance from the Earth to the Moon and ES is the distance from the Earth to the Sun. EM/ES is approximately 1/390 (=238000 miles/93 million miles), so $\alpha$ works out to be 89.85 degrees.

Aristarchus was trying to do the reverse calculation to find the distances. If the angle $\alpha$ could be measured accurately, the relative distances EM/ES could be calculated using the same formula. The problem is the difference between 89.85 degrees (the Sun is 390 times farther than the Moon) and 90 degrees (the Sun is infinitely farther than the Moon) is a very small angle and difficult to measure exactly. If the Sun were a lot closer, then $\alpha$ would be a smaller value, and the error in the measured angle of $\alpha$ would be restrict the value of EM/ES to a more meaningful range than "infinity".