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but to be a good CEO or CTO you should distinguish "their" and "there". And do you have only one customer who faces invoices? I suspect you meant something rather different by that sentence.

Agree with john.a.wills. Most organisations would place this type of accounability in the "Chief something" space. As an Account Chief Engineer for a large global tech company, my role often crosses into the domains you mention despite our having

Not much of a CE. Couldn't get the EDIT button to work after posting an unfinished post! To recap:
Agree with john.a.wills.
Traditionally the IT industry consists of highly skilled individuals who are (naturally) very proud of their traditional titles. Storage engineer, backup engineer, network consultant, security consultant, sys admin. As we take on responsibility & move up the hierarchy, our title is often less clear.
Most organisations place your accountabilities as "Chief something".
* CTO is close, only a CTO is Account Resource focused, not HR/People Mgmt (you listed Team Leader).
* As Account Chief Engineer for a global tech company, I often cross into your domains despite there being assigned resources with the titles listed! But this doesn’t make it right!
* Chief Architect doesn't suit. An Architect takes a concept to reference architecture to high level design. Architects don't like to get involved with Delivery.
* A Solution/Tech Designer is often tower or project specific - too limiting.
* Tech Consultant – Pre Sales? Maybe, but like CTO a TC is not concerned with HR/People Mgmt issues.
Keeping with tradition, your title options are probably Chief Technology Officer/CTO, or better still Chief Information Officer/CIO. In any case, a title is just a title & my view is to use the mismatch of responsibility as an opportunity to further develop your skills. Let me demonstrate...
Like most females, I like to shop for jewellery. When I ask for say, "...a pair of yellow gold huggie earrings", most stores show me the same boring products. What I'm really looking for is something unique that no other female on the planet will ever own. A shop assistant who can clearly identify the customers ACTUAL requirements & back it up with a product or ideally, an ENTIRE SOLUTION (earrings, pendant, chain, ring, bracelet), will most certainly win the sale!

The technology industry business requirements have shifted. Employers no longer need people with a title, they need people with an edge. Multi-skilled resources are the order of the day. Single domain SMEs are still highly respected & sought after, for specific roles. To stay in the game we need a flexible, scalable, unique edge & present it to the right people at the right time.
What is my Job Title?
Pick a title, any title (I like IT Consultant because it's so vague) and concentrate on building a broad repertoire of sought after skills. And when you find a fitting new age title, please share :)