I Comparing Earth's diameter with Moon's diameter

The sun’s diameter is about 400 times larger than that of the moon – and the sun is also about 400 times farther from Earth. So the sun and moon appear nearly the same size as seen from Earth.

the size of a celestial body is usually measured by the angle subtented at the observers eye. and the angle /angular diameter depends on the distance at which the body is placed.....as an example the above statement during solar ecclipse the disc of Sun and Moon appear to be of the same size.
if you see satellite photographs- the sizes vary as the space telescopes position varies.
if you wish to see exactly the calculations of Aristarchus ...you can visit the site (many a time his geometry gave wrong figures but later got corrected) he could draw triangles in space between celestial bodies and gave arguments/calculation of sizes.

http://www.astro.cornell.edu/academics/courses/astro201/aristarchus.htm

 

You are confusing two entirely different events. In a Solar eclipse, the Moon subtends the same (approx) angle as the Sun - and blanks it out. In a Lunar eclipse. Earth is a lot bigger than Moon so the eclipse (Large Earth blocks Sun from small Moon) lasts for quite a long time and is seen from all over the Earth (the parts facing the Moon at the time). In a Solar Eclipse, the Moon blocks the sun and has to be 'exactly' right for a total eclipse (same angles subtended). The Moon's shadow only covers part of the Earth's surface at any one time. It sweeps over the earth's surface
Total Lunar eclipses are more common than total solar eclipse because of the relative sizes and consequent different required precision.