When learning alternate picking, I have heard two definitions. One is to pick up and down strokes strictly alternating with each stroke for each note and another is to pick down on downbeats and up on upbeats. What are the rules for alternate picking?

How would you pick something like this riff in Metallica's One?

D|---------------2-----------------2----------------2----------------2--|
A|---------------2-----------------2----------------2----------------2--|
E|-0-0-0-0-0-0----0-0-0-0-0-0----0-0-0-0-0-0----0-0-0-0-0-0-----|

Would it be D-U-D-U-D-U-D
then U-D-U-D-U-D-U
then D-U-D-U-D-U-D
then D-U-D-U-D-U-D

The low E notes are in two groups of triplets.
The second definition will help you stay in time but it doesn't really matter in the long run as long as you're practicing consistently.

For this riff, I play DUDUDUD and repeat - Starting every set of sextuplets with a downstroke is easier for me (and I presume most people).
Quote by sweetdude3000
When learning alternate picking, I have heard two definitions. One is to pick up and down strokes strictly alternating with each stroke for each note and another is to pick down on downbeats and up on upbeats. What are the rules for alternate picking?

The bolded definition is the correct one.
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Quote by Geldin
The bolded definition is the correct one.

Given that, Would it be better practice in the long run to begin each double stop with a D and then U then D and then U et cetera to get the uniform U D U D ... for each note? I am just wanting to know what's best so I don't end up practicing the not so efficient way
If you're playing quarter, 8th, or 16th notes in 4/4, you should hit every downbeat with a downstroke.

If you're playing 8ths and 16ths in 6/8 (or 12/8, 3/8, etc), or triplets/sextuplets in 4, you'll alternate up/down.

That all assumes you're just rocking continuous picking. The general rule of thumb is to keep your downstrokes on the strong beats unless you have a good reason to do otherwise.
Quote by sweetdude3000
Given that, Would it be better practice in the long run to begin each double stop with a D and then U then D and then U et cetera to get the uniform U D U D ... for each note? I am just wanting to know what's best so I don't end up practicing the not so efficient way

The way I play that passage, I start the sextuplets with a down stroke, which lets me use a down stroke on the power chord at the end, then start on a down stroke again on the next set of sextuplets, and so on.
I have an album. Listen to it.

The album is streaming here
Quote by sweetdude3000
Given that, Would it be better practice in the long run to begin each double stop with a D and then U then D and then U et cetera to get the uniform U D U D ... for each note? I am just wanting to know what's best so I don't end up practicing the not so efficient way

Well, if you do a strict D U D U pattern, beginning the next set of semiquavers with an upstroke, you will have to play the second chord with an upstroke. That gives quite a different sound to downpicking it. Unless your upstroke is as good as your downstroke (doubtful, I've never heard anyone with an equal strength upstroke and downstroke), then it'll be best to downpick on the beat, which will accentuate the beat and give it that heavy feel the piece has.
When learning alternate picking, I have heard two definitions. One is to pick up and down strokes strictly alternating with each stroke for each note and another is to pick down on downbeats and up on upbeats. What are the rules for alternate picking?

Alternate picking is alternating between up and downstrokes, but in practice, 99% of the time you want your downstrokes on the beat and upstrokes on the offbeat, which helps a lot with timekeeping and correct accents.