Are there publicly agreed-on standards or definitions of what constitutes peer review?

No, there isn't. Different journals/publishers will have different guidelines.

Here're some examples of publicly-published reviewer guidelines:

• Wiley
• Elsevier
• PLOS One

To illustrate the differences, Wiley's guidelines say "What is the main question addressed by the research? Is it relevant and interesting?" (emphasis mine). Meanwhile PLOS One doesn't care about "interesting". To quote a testimonial: "I published with PLOS ONE because I love the idea that acceptance is based on quality of the work, not whether it's trendy or important in the eyes of a few editors."

If there were a strict definition and satisfying it became important, then the disreputable journals would find a way to game that.

It's straightforward to determine whether a journal is reputable: ask the experts (the word reputable, after all, refers to reputation). Experts here means, for instance, faculty working in the relevant field at reputable universities. I think the average person has a good idea of what reputable universities are.

is there a commonly-agreed-on definition of what this means?

Generally, good peer-review means that experts in your field carefully read your work and evaluate its merit. The definitions of "experts" and "carefully" vary widely across disciplines. Some fields require several rounds of meticulous back-and-forth between author and journal; others are less stringent.

Is there a publicly verifiable process to verify whether a journal is peer-reviewed?

There are several journal rankings, but I think your best bet is to look at the editorial boards/published works. Are they reputable researchers who are considered `pillars' of their respective fields? Are the papers that were published in that journal of general good quality, and relevant to your prospective submission?

Some red flags to look out for when deciding whether a journal has good practices:

• Journal is publicly listed on some blacklist (such as Bealle's list)
• Journal charges exorbitant fees for publication, and promises a fast turnaround time on publications.
• Journal emails you: good, reputable journals don't need to spam researchers for articles. That said, sometimes this is valid (say, you published in a CS conference that has a direct-to-journal track for fast publication turnaround, that is ok).
• Editors are relatively unknown people from non-research oriented universities
• The papers submitted there are garbage (not 'unclear' but absolutely unreadable/weird)

In general, research publications are about disseminating your ideas to a certain community. It's important to see where your community publishes, and go for these venues.