OK, I’m honestly not sure if this is a valid answer because I don’t know what the question is any more, but some quotes:
Actual, Published Rules
If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. You suffer a -6 penalty with your regular attack or attacks with your primary hand and a -10 penalty to the attack with your off hand when you fight this way.
“This way” refers to the way in which you get an extra attack per round. It is optional both to take a penalty and to gain the extra attack; you do not take the penalty unless you choose to do so and if you choose to do so, you get an extra attack.
This just repeats some of the rules from the Combat section also titled Two-Weapon Fighting.
Rules of the Game
First of all, I want to emphasize that Skip Williams often misspoke, occasionally quite blatantly, during these articles. They do not carry any official rules weight, and any contradictions between these articles and the official rules should be completely ignored: the official rules trump these articles every time. See also the errata rules for 3.5, which are printed at the top of each errata document.
Off Hand, Off-Hand Weapon: When attacking with two weapons, the character must designate one of his hands as his off hand; the weapon held in that hand is treated as his off-hand weapon. The game rules don’t really care about whether you’re right-handed or left-handed, and it’s even OK to change your off hand designation from one round to the next.
Right off the bat, even using the Rules of the Game articles that have no bearing on the actual rules, these terms apply only when attacking with two weapons. Not when you attack with a weapon while holding another in the other hand, not while you attack with a weapon while holding nothing in the other hand, not when you attack with a weapon while having a blade sticking out of your boot, but only when you attack with two weapons simultaneously, i.e. use the Two Weapon Fighting combat option to gain an extra attack.
Primary Hand, Primary Weapon: If a character is only attacking with a single weapon, it’s fine to treat that as a primary weapon, regardless of what hand it’s held in. When a character fights with two weapons, he can designate either one as his primary weapon.
To belabor a point, while you aren’t using two weapons, whatever weapon you are using is the primary weapon, period. You don’t even have an off hand or an off-hand weapon.
Once again, you can carry a weapon without using it, which means just because you have a weapon in the other hand (say, a shield, or an unarmed strike), does not mean you are using Two Weapon Fighting and therefore you get no penalties.
You do not have to choose between the attack and full attack actions until after you have made your first attack on your turn (see page 143 in the Player's Handbook). However, if you intend to attack with two weapons during your action, you must take the correct penalty for each attack or give up your opportunity to use your second weapon (because the rules require you to take a penalty on attacks you make with both your primary and off hands).
(emphasis mine, though the italicized Player's Handbook is present in the original)
Once again, the choice to take the penalties associated with Two Weapon Fighting are conditional on choosing to attack with two weapons, i.e. use Two Weapon Fighting. He explicitly states that you can choose to take no penalties, you just don’t get the extra attack. However, Skip here implies that the torch is unavailable even if you have attacks for other reasons (high BAB, haste spell, etc), but if that implication is intentional (rather than just an artifact of assuming level 1 without special features as a base case), then he is wrong. Yes, he can be wrong, and if he is, his inaccurate statements have absolutely no bearing on the actual rules: the Rules of the Game articles are intended only as clarification, and if mistakes were made, they do not serve as errata, changing the rules. The errata rules make this very clear. The Player's Handbook is the primary source on all of this, and anything contradicting the PHB should be ignored, under the rules.
Once you take a two-weapon fighting penalty, the penalty applies to all the attacks you make with that hand during your current action. It does not apply to attacks you make during some other character's turn. For example, say your torch-wielding swordfighter from the previous example has a base attack bonus of +10. With the full attack action, the character could make two attacks with the sword: one at +10 and the other at +5.
Here we see how ephemeral the "off-hand" designation is: it lasts only for the full attack in which the Two Weapon Fighting option is used. Attacks thereafter (an attack of opportunity in the same round, any attacks at all in future rounds) do not receive any penalties (unless you use the Two Weapon Fighting option again). You do not have a permanent off hand, you merely have an off-hand weapon for the duration of the full attack in which you choose to use Two Weapon Fighting.
Ranged weapons you don't throw use most of the rules for two-weapon fighting; however, the Two-Weapon Fighting feat and related feats don't apply to ranged weapons that aren't thrown.
Just another example of Skip being wrong. This rule is not present anywhere in the actual published rules, it's literally something he appears to be making up for the purposes of this article. Under the official rules, this should be ignored as an error.
Mentioned only for completeness, this article does not discuss off-handedness in any meaningful way that wasn't already covered in previous articles.
Just as another example of Skip being wrong, though this one is much more dubious:
*The damage bonus for the double weapon's primary end is +3 Strength, +2 enhancement, +2 weapon specialization. The damage bonus for the double weapon's off-hand end is +1 Strength (1/2 of +3) and +2 weapon specialization.
The rules do not actually state at any point that double weapons are treated as a one-handed weapon and a light weapon for the purposes of damage. They state they behave like a one-handed weapon and a light weapon for the purposes of attack rolls but say nothing about damage rolls. By a strict reading of the rules, you actually get 1½ Strength to both ends. This is likely an oversight, however, so this particular “mistake” may be more with the PHB's editors than with Skip. Still, if the goal was to clarify the rules as written in the PHB (which is nominally the purpose of the articles, and certainly the only valid thing for them to do under the errata rules), then this is another error.
You only have an off hand, of any kind, while using the Two Weapon Fighting combat option. Which hand is “off” is freely chosen at the time you choose to invoke the Two Weapon Fighting option, and that designation applies only during the full attack in which you are using the Two Weapon Fighting option. If you do not use Two Weapon Fighting (i.e. do not get an extra attack), you do not have an off hand at all. There is no innate property of either hand that makes it an off hand. All 3.5 characters are, effectively, ambidextrous.
It is only when you choose to attack with both weapons simultaneously (i.e. use the Two Weapon Fighting combat option) that you must designate an off hand; this is the hand that you do not favor in the attack; you put your body and weight primarily behind the primary attack, and off hand is just a little extra (hence halved Strength). It is a function of the particular action you are undertaking, not a property of that arm, hand, or weapon.
The Glossary exists only as a secondary rules source, by definition. The Glossary exists without context: the statements contained therein are only for easy reference, not rules in and of themselves. By the errata rules, in the case of any contradiction between the Glossary (a secondary source) and the original place in which the term is defined and used (the primary source), the primary source trumps the secondary source. The Glossary can only repeat information, not provide new information; that would be a contradiction. In this case, there is no contradiction: the Glossary merely provides a definition, but without context. By looking to the primary source, the context becomes clear: the term “off hand” only applies while using the Two Weapon Fighting option to gain an extra attack.