I constantly get asked about this topic. That's usually a clue that there are many different teaching methods/explanations floating around. It also means that there is probably no one "exact" right answer.  I agree with all of this. 

So, the best I can do is give my opinions and discuss the situation in general.

It is true that 6-5 hands should "come alive."  The shape alone makes the hand worth much more than its HCP.

But, the real problem is which suit to bid, how many times, and in what order.

6 cards in a MAJOR and only 5 in the minor

The opening bid decision is easy. Start with the MAJOR. For example, open 1 with each of these:

3
AQJ765
KQ762
2
, A2
K108765
AJ765
--
, --
QJ8654
AKQ32
65
.

Open 1 with:

AJ7654
3
2
AQ876
 or J108765
A2
--
AK652
.

After partner responds, should you rebid the major or introduce the minor? There are two schools of thought:

1) Always introduce the second suit on your second turn

2) Introduce the second suit only with extra values

I'm not a fan of either rule. I like to evaluate it on a hand-by-hand basis. The relative strength of the two suits is important. With AKQJxx in the major and Jxxxx in the minor, I'd be likely to rebid in the major.  Unfortunately, there is no universal or exact answer to this rebid conundrum (probably why I get asked about it so often).

6 cards in a minor and only 5 cards in the MAJOR

Here, it isn't clear which suit to open. The order and quality of the suits is relevant. Opener must consider his rebid problem. With clubs and spades, it is usually okay to open 1, because it is convenient to get the spades in next. But, when the major is hearts, you risk losing the heart suit (because a heart rebid will often be a reverse). Here are some examples:

 

AQJ74
32
--
AK8765
This is a comfortable 1 opening. You likely will find it convenient to rebid in spades (twice).

 32  
 AQ1074  
 --  
 AK8765
 

This one is tougher. If you open 1 and partner responds 1, you'll be able to easily bid hearts now and then again later. But, if partner responds 1 or 1NT, you would have to reverse to show the hearts. Are you worth it? Maybe. This is quite a good hand. So, 1 is acceptable (yet not 100% clear).  Now, make it a little worse:

 --  
 AQ874  
 65  
 AJ8765
 

A 1 opening feels best. If you open 1, you risk losing the heart suit. This hand, with only 11 HCP and a spade misfit is nowhere near worth a reverse after 1-Pass-1.  I'd like to get the hearts in with a 1 opening and bid clubs next (several times if possible).

6 HEARTS and 5 SPADES

If you open 1 and partner responds 1NT, a 2 rebid would be a reverse, showing a strong hand. So, if you were too open 1 with, say:

 AJ765  
 AQ7652  
 4  
 6  and partner responded 1NT, you would have to bid 2 next (supressing the 5-card spade suit and possibly missing a 5-3 spade fit). For that reason, you could open this hand 1 and bid hearts at the next turn and the next turn and the next turn ad nauseum.  With more strength, maybe

 AQJ76  
 AKJ765  
 43  
 --
 , you could open 1 and would have enough strength to reverse into 2 after a 1NT response (and you would bid spades on your third turn as well).

Other issues:

> With 5-5- in the black suits, I prefer a 1 opening, but don't feel strongly about it.

> Sometimes I will open a weak 2-bid with 6-5. If the hand isn't worth opening bid strength, why not a weak-2? For example, I'd gladly open 2 with: 3
AJ10876
Q6543
2
. I can't see any reason not to.

Summary:

There is no "one size fits all" answer for how to bid 6-5 hands.