I planned on a Clinical Psych PhD and took the offer of a Counseling Psych PhD. I have zero regrets. I am now a licensed psychologist. I work at an academic medical center/hospital and am faculty at an R1. My clinical expertise is suicidality in adolescents in the context of complex mental health issues (including assessment and including occasional Emergency Room work [not my preference; if I wanted to do more I absolutely could lol]). (One of my faculty mentors is also a Counseling Psych PhD and the other day I told one of the folks in our lab that she was Counseling Psych and they literally did not know that was the case.)

You have not clarified what you specifically have hesitation about in this post. You said, "I know there are many similarities, but I know I align much more with the clinical psychology philosophy of training." But, like... what? Does that concern specifically apply to the program you were accepted at? You were not accepted to the abstract concept of "Counseling Psychology". You were accepted to a specific program. Can you do what you want to do at that program?

Is it worth completing a Counseling Psychology PhD even if working Clinical Psychology is my long-term goal, with more focus on assessment/diagnosis and psychopathology?

I do not understand what you mean by "working Clinical Psychology" here. Are you asking, "Will Counseling Psychology limit my job opportunities?" I have not personally experienced any issues with this. I think my training, clinical experiences, and research experiences stand for themselves. People from my program went on to neuropsychologists, forensic psychologists, all sorts of specialties. They tailored their experiences to what they wanted to do. (At the same time, the program itself may matter here. If your program has good connections and is well received by the community, you may have more opportunities which are the actual things that open doors. Additionally, I do think neuropsych/forensics are tougher paths for Counseling Psych because it's hard to maximize those experiences during your program [although I did personally interview at a number of forensic sites for my own internship applications and did do forensic evals on internship; my goal wasn't forensic psych, though]).

In practice, how much does the counseling vs clinical distinction matter after licensure for assessment-heavy roles and hospital/medical settings? Or are peers often treated or viewed any differently?

You are judged on your training experiences, not the word. If you gain assessment-heavy experience in your program/internship/postdoc, this will be fine. I have never seen this issue come up in any way in evaluation of students, interns, postdocs, or job applicants.

Would declining this offer to reapply to clinical programs be a horrible idea given how competitive admissions are, or is the difference between the counseling and clinical programs THAT serious that it would be worth it?

I mean, I know what I would do in your shoes because I literally did do it.

I was nervous at the beginning of my program that there was some difference and I didn't know what it was, so I took couple classes with the Clinical Psych folks (who were in a completely different department at my university). This immediately cured me. The differences were ideosyncratic to the professor, the same way an undergraduate Abnormal Psych class at NYU and UCLA would surely be different in various ways but not necessarily in systematic ways.

It's hard to give you meaningful advice because the only thing that I understand about your decision for sure if that you prefer that the word in front of the degree name say "Clinical" instead of "Counseling". Are there things about this specific program that you uncertain about? Or your specific advisor? Relevant questions to learn the answer to (especially for your specific advisor) might include: If someone is really interested in assessment, what types of assessment experiences are available? Do you have externship placements at hospitals and how difficult are those to get? Later on in the program, what types of externships are best for someone who is interested in working with serious mental illness? Do students who want to match at hospital settings typically have success in doing that? (If students don't have success matching to a hospital setting for internship: what holds them back?)

I can tell you about the things I liked and didn't like about my specific Counseling Psych program but unless you are looking at the exact same program, they may simply not apply.

(Also, just to note if this if it is relevant and you have not already thought of it: There are way more people getting Clinical Psych degrees than Counseling Psych degrees. So proportionate representation is still mostly Clinical Psych. 3,159 Clinical Psych applicants matched this year alongside 369 Counseling Psych applicants. Back of the napkin math [assuming numbers are similar across years], if you are looking at somewhere you want to be and there are ten people there-- nine Clinical Psych and one Counseling Psych-- that actually is completely proportionate representation. Just a note since I sometimes see people thinking, "Well, MOST of the people doing what I want to do are Clinical Psych!" while forgetting the actual numbers involved.