A new moon is also when the Earth is in between Sun and Moon.
You got this one wrong. The description above applies to full moon. Correct is:
A new moon is when the Moon is (roughly) in between Sun and Earth. Then we look at the side of Moon that doesn't get sunlight, making Moon appear very dark (invisible to the human eye).
- New moon is Sun - Moon - Earth.
- Full moon is Sun - Earth - Moon.
At new moon, solar eclipses can happen if Moon gets located exactly between Sun and Earth, thus blocking the view from Earth to Sun. This only rarely happens because Moon's orbit around Earth is tilted a bit against Earth's orbit around Sun, meaning that in the "new moon" situation, most of the time Moon passes a bit "above" or "below" the Sun.
Similar for full moon and lunar eclipses. Full moon happens when, looking from Earth, Moon is opposite to Sun (Earth roughly between Sun and Moon). Then we look at Moon's side that gets full sunlight. Only if Earth is exactly between Sun and Moon, then Moon enters into the shadow cast by Earth, thus becoming very dark - what we call a lunar eclipse.