When you encounter a designer’s profile and assets somewhere, what helps you differentiate a mid-level from a senior designer? what about an experienced individual contributor from a design manager?
Typical 'calling cards' for more senior digital° designers include the following:
For their website: Limited amounts of work, a very focused delivery mechanism
(as opposed to surfacing 20+ projects, a sprawling website, 10+ social links, etc.)
Written (and appropriately detailed) case studies, supplied with appropriately-scoped visual helpers
People who have worked in the 'Freelance' space for extended amounts of time — I've heard it repeated often that folks who can swing the freelance lifestyle and do well are able to manage various aspects of the 'design job' that translate to a Senior skill set.
An ability to direct and shape the 'culture' of a space — this is a bit more personal of an opinion, but it's clear to me that people who are able to manipulate immaterial spaces like conversations, zeitgeists, and organizational culture are able to operate in 'higher' spaces with regards to designed forms, which I personally find highly valuable in people I work with.
This is not an exhaustive list, just some things that popped into my head as of this writing. I admittedly can't offer up specific insights/opinions on how one might discern a 'really good' product manager, but I imagine there's much more overlap than people would like to admit.
*I mention digital because the tells for 'seniority' vary across the many and varied design disciplines — some excellent design can't be conveyed via imagery, or detailed documentation, or writing, etc.

How does the “type of work” they’ve done come into consideration when you’re hiring? do the different role titles in the design field affect this? does industry?
For me, personally, no. I'm a strong believer in the ability for baseline 'good design practice' to translate across disciplines. Pragmatically, and for Tumblr specifically, this isn't the case, and we often constrain designers we hire into people who have worked on 'mobile product' before, self-identify as digital product designers, and have a stronger than not visual design background. I don't believe in this per se.
Generally, I think the industry writ large operates in a similar way to Tumblr, privileging very specific skillsets that are often constrained and limited to the types of tooling one uses. This is such a young industry that I'm of the opinion that most 'best practices' regarding hiring are fraudulent and short-sighted at worst, limiting potential at the very least.
If you have any comments in particular on how you make an initial determination on whether a product manager might be worth talking to, I’d love to hear that
A general heuristic I like to go by:
If I could hire the person as a capable classical User Experience designer, they'd probably fit really well as a product manager. The overlap between the two disciplines is large imo, with simply different weighings in priority given to the lifespan of a project.

Edouard on Product Design