The Elementalist Fighter class is more powerful at every level than a standard Battlemaster Fighter.

I'm using the Battlemaster Archetype as a reference point as the OP has stated that it is an influence on the homebrew class design and because the class features closely mirror that of the Battlemaster.

TLDR; Elementalist Fighter blends almost all of the features of a Battlemaster, adds a sprinkle of Barbarian and then adds a tonne of powerful abilities to boot.

• Second Wind comes 2 levels later, decreasing survivability at 1st and 2nd level, before returning to the status quo at 3rd.
• Weapon Bond is almost identical to the Eldritch Knight feature and comes 2 levels early. It somewhat undermines one of the EK's unique features but is not too major.
• Imbue Weapon is a reskinned Barbarian Rage, the quintessential Barbarian feature. It is more offensive focused than defensive. There doesn't appear to be an end condition, but I'll assume it's the same as Rage. Almost everything here feels out of flavour for an elemental focused fighter. Psychic resistance, temporary darkvision, changing the advantage on checks and saves to Charisma, and advantage on saves against being charmed are not, in my opinion, things I would associate with a class called "Elementalist Fighter". Mechanically, choosing the additional damage's type is strong as it somewhat enables low-level melee fighters to circumvent physical damage resistances, one of their greatest banes. It is somewhat comparable to the 3rd level Elemental Weapon spell, minus the to hit bonus and magical weapon consideration. Losing the physical damage resistance of Rage is not a suitable drawback, as Fighters are much more capable of achieving high ACs at low levels than Barbarians, through heavy armour, shields, and fighting styles. So the loss of defensive abilities is negated by the classes intrinsic high AC and the damage capabilities are greater than the standard Rage, with some odd extras thrown in.
• Maneuvers. Having extra restrictions on maneuvers is sure to only cause confusion. The save DC being Charisma based is also quite unexpected and again outside of the classes flavour. Some of the new maneuvers are far above the current power level. Elemental Slam stands out as particularly egregious. Fighters lack AoE as a balance requirement. Giving them access to AoE which does good damage for a relatively low resource cost, is definitely going to cause them to be overpowered relative to another fighter of the same level. In conjunction with the powerful secondary abilities that many of the damage type Slams offer, I don't know why you wouldn't pick this maneuver over any other.
• Ancient Blade doesn't have any real balance concerns. It does, however, cause a great deal more work for your DM, which is something that I would consider poor design. Know your Enemy of the Battlemaster is pretty whatever, but it doesn't cause any additional preparation or forethought that could be used on the game as a whole, instead of one character.
• Elemental Incarnation is absolutely insane. Indomitable is already a strong class feature, and EI blows it out of the water. It evaluates to casting one of the Investitures of X spells, without the need for concentration, once per day, at the cost of one level of exhaustion. And this is in a non-casting class, 2 levels earlier than a full caster could. The sheer damage output and/or utility that this feature could provide in its 10 minutes is incredible. Especially considering most conditions that require a save also give one each turn. Potentially missing a turn or two is well worth this.
• Elemental Explosion is again powerful because it gives Fighters access to powerful AoE at a relatively low cost.

If given the choice to play a Battlemaster or an Elementalist Fighter, the Elementalist is the clear winner. With its access to cheap AoE and additional damage output, it does everything a Battlemaster does and more. It suffers from a lack of identity, borrowing iconic features from other classes, and also having a mix of features that don't resound with the classes core concept.

Recommended balance changes to Noble Knight

Yes, a name change is one of the suggestions for thematic reasons. 😊

1. Change the name to Noble Knight.
   Why? Based on our comments discussion (and you searching for a term that fits in not-your-native-tongue) Noble Knight almost exactly captures the overall theme of your medieval knight concept, while "Sir Knight" is more a form of address.

2. Change the As Good as My Word Cultural Implications entry to read:

Cultural implications of your view as a Noble Knight are widely known. Starting at 2nd level, you have advantage on Persuasion and Intimidation skill checks when dealing with creatures of humanoid who come from races or cultures where knighthood left a significant imprint.

Why? This is going to vary by campaign, so giving example races is a case of both too much specificity and not enough. That's a tidier way to say what you are trying to say.

3. Your 7th level feature is overfilled with features.
   a. Peripheral Vision is overly loaded with features. Suggest you remove the "you can also" feature, and make it an always on feature. The ignoring of half and three quarter cover doesn't seem to fit the feature.

4. Multiclassing - change the prerequisites to this:

Prerequisites. To qualify for multiclassing into the Sir Knight class, you must meet these prerequisites: 13 Strength or 13 Dexterity

Why? Be consistent with how MC works in this edition.

5. Tools for the Job
   a. First off, keep the cantrips. Nice thematic touch.
   b. Suggest that either these can be cast as a ritual once per long rest, or, you cast any of them a number of times equal to proficiency bonus per long rest and not as a ritual.

Why? That's a pile of spells to get all at once, and, as a point of comparison, the Barbarian gets a similar ability (to cast two utility spells as a ritual) as a Totem Warrior at third level. Since you link a lot of stuff to proficiency bonus, I suggest that you increase how many spells you can cast per long rest by indexing to proficiency bonus. (At level 5, for example, 3 rather than 2). This will reflect the "get better as you go up in level" concept that you generally have.

6. As to And the Pyres will Burn

Starting at 6th level, when you cast Create Bonfire, the spell creates a magic bonfire that fills a 10-foot cube instead. Additionally, you can use your bonus action on your turn to move the bonfire to another location within range.

Your teleporting bonfire, up to 60 feet per round, is more mobile than Flaming Sphere. You need to restrict the distance that you can move it per round to 30' or even less.
Here is "why" the "even less" is my recommendation. This feature as written is better than 2d level spell flaming sphere. Since you are using a class feature, it being a little more powerful is OK, but, because it does more damage than flaming sphere I'll recommend that it's movement be limited to 10 or 15'. From a cinematic PoV, the bonfire slides rather than teleports, or rolls over, so having a 10' bonfire roll 10' 'looks' more like a moving flame. Yes, this suggestion is an appeal to both balance and verisimilitude. Other spells that have a moving AoE, like Moonbeam, also inform this suggestion.

7. Call to Arms. This is too fiddly. Remove the following and it fits better.

If a creature loses all those temporary hit points, it instead can add your Charisma modifier to its saving throw for the remaining duration.

8. The Last Obstacle
   Change the HP gives healing to adding Temp HP to be consistent.

9. Weight Control: I had originally assessed this as a possible means to Dual Wield great axes and two handed swords but your comment allayed that concern. The bonus to versatile weapons (war hammer/longsword) is nice but not over the top.

Overall evaluation

As with a lot of home brew, the temptation to add too many features is hard to resist. It was a good idea to get a scrub from other perspectives. My only significant concern is the Weight Control feature since it overwrites a core game mechanical aspect of melee combat.

These are some up front changes I'd make to keep the balance point closer to somewhere between Paladin and Fighter. Overall, I would need to play test the Inquisitor option a bit to see what might break, but the Commander seems to fit within the balance box if you make the corrections I suggested above.