Maarten thought I'd have some good perspective to add here as that's exactly what I do - coach Product Managers/Owners/leaders and product teams - and I'm not alone either. There are a growing number of people and places who are doing so for the exact reasons you've outlined.
What I've been witnessing over the past ~3 years are that many places have begun to realise that a large percentage of Scrum Masters/Agile coaches have little experience in Product Management or in Discovery/Design Thinking. And they're realising that you need both to build successful products.
In some cases they hire both, agile experts and UX experts but often these people end up at odds with each other - I've witnessed it first hand, designers arguing with agile coaches to validate before developing and agile coaches labelling discovery as BDUF.
Worse because they don't understand Product Management and validation through anything other than shipping software, it adds to the proliferation of feature factories and continues to dilute the PO role to be the tactical role that is widely accepted as today in the market.
But the good news like I said is that organisations (at least what I've been observing) have started to realise the value in having people with REAL product experience. People who know what it's like to be in the trenches talking to customers, who has turned insights into opportunities, shipped products and drove customer outcomes - not an expert in stabilizing a teams sprint velocity.
As such I'm seeing "product orientated" coaching roles become more common - in fact I've done 2 transformations now which were post an agile transformation and it was to help them become a "product-led" company.
They had agile people in there 2-3 years prior but they were still very much a feature factory. It wasn't until they hired a new CPO that she hire us and we helped her change that, helped with product stratgey, upskill the POs, hire a whole load more designers and org design to be more of a product-org structure with logical product groups and introduce continual discovery.
So I totally get I haven't directly answered your question. But I thought the background would be pertinent and writing this now I think I might turn it into a future blog post ;)
On your question of how to differentiate yourself well I've witnessed in particular that companies in Australia/NZ part of the world start to use titles like "Product Coach" "Product Manager Coach" and "Product Delivery Coach".
It's not just here it's also getting traction in Europe from what I can see. Seb (a fellow ex-ustwo peer in London) wrote this at the beginning of last year about the "Product Coaching" work that he is doing at Founders Factory in London.
So one part to differentiate yourself is to be explicit about it - put it in your title.
And you don't need to complete drop the agile part if you don't want to, you'll still coach agile to a degree - afterall agile is a large part of how we do product! And I currently lead a team which has both Product coaches and agile coaches in it (cross-functional team if you would).
But the key difference to me between agile coaching and product coaching is best surmised in this quote by Jeff Patton (who remember has his origins in UX and Product Management): "There are consultants and consultancies that competently specialize in training and coaching in any of these areas [agile, lean & ux], but that’ll leave your organization with the challenge of integrating them because, you need all three to predictably build successful products. That’s where I and the partners I work with come in."
Or as he also states "we're the glue that connects good product management and strategy, lean user experience and agile delivery practices together."
That's the difference to me. I've worked with very few agile coaches who can do that - combine lean, design thinking, product management and strategy with agile.
Hope that helps, or at least is some food for thought. Happy to connect if you want and chat deeper on this topic - https://www.linkedin.com/in/ant-murphy/