I like this! I tend to be very picky with other homebrew but I quite enjoy this. I've been of the opinion for a while that more fighter features should proc off of second wind and action surge. I especially love that you referenced both indomitable and inspiration, which I feel are often overlooked. Good job!

I wrote a full Commander class with the same vibe a while ago because I enjoy the trope but don't like the other warlords out there (kibble's is my least favorite. I also dislike warlord as a class name but that's just me). But I'd play this

Update v2

• Changed the names of two features (Inspiring Leader -> Swift Strategist, Commanding Presence -> Dominating Stature) as they are already used for other existing features.
• Nerf: Changed Vanguard to only affect saving throws against the frightened condition (instead of charmed and frightened) and only if you're not incapacitated.

Update v3

• Buff: Rallying Cry gives a bonus to attack and damage rolls equal to the Purple Dragon's Charisma modifier.
• Buff: The upgrade to target two allies with Action Surge has been moved to 15th level (was previously 18th).
• Buff: A new feature for 18th level lets the Banneret protect itself and its allies from incoming attacks.

This will probably be the final version of the subclass, as included in my Tavern Tales compendium. Click on the link to read the wondrous item related to the subclass, as well.

Looks good! The only point of critique (other than features in 5e not usually interacting with the inspiration mechanic, but that seems to be changing for 5.5e) is that Inspiring Leader is already the name of a feat, and Commanding Presence is the name of a Battlemaster Manoeuvre (which can be gained through a fighting style or a feat). This could cause a character to have two features with the same name on their sheet, particularly because those abilities play to the same theme (charismatic leader). I don't know what I would change the names to, though.