The rule is "е" sounds like "ye" after a vowel, does the е[ye] have to be right in front of the vowel to sound like [ye] or can it be anywhere behind "e" throughout the word?
You are right. If
e is preceded by a vowel, it is read as
моет, стирает, убирает, etc.
We will also read
je if a words starts with the letter:
енот, ещё, если, etc.
e is preceded by
ь, we shall read it as
je as well:
подъезд, съезд, въезд, варенье, etc.
However, in words like
e is often reduced to
моит), and words like
ещё (see the second rule above) may be read as
если will always be
e is stressed). The reason is it's just simpler to pronounce the words like that. Technically, if you read
je in those words, you'll be right, and no one will ever correct you. Many people will not even catch the difference. Anyway, it's important to note the reduction.
Better put it this way:
еimmediately follows a consonant, no [j] is added:
привет, тебе, же.
Otherwise it gets a [j] sound in front of it:
ел, поел, съел, колье.
Some words have both type 1 and type 2
еле, веер, белеет, змееед.
Same rule applies to
я, ё, ю.
Minimal pairs to practise the distinction:
полёт/польёт (flight/will pour),
полю/полью (I weed/I will water),
чего/чьего (what/whose), перо/Пьеро (feather/Pierrot).