What your instructor was probably trying to get at is called PDPIC time. Performing the duties of pilot in command. THIS IS NOT PIC TIME! It is simply an endorsement to satisfy the FAA requirements of "Solo" cross country or "solo" time in a multi. However, your instructor is always going to be with you. I usually make an column labeled PDPIC or have an asterisk next to every PIC time logged.

On the 8710 you log it as PIC and solo time, however your log book should not match this. Because in the FAA's eyes this is technically "solo" time.

You can not log PIC in the aircraft unless you are rated or sole occupant. Which in your case, is neither of those.

THIS^

all of my "solo ME PIC" time without a rating was specifically labeled as such in my logbook.

Yes, If I understand correctly, you're wrong. If you are not the sole occupant of the aircraft, you need to be rated to log PIC as the sole manipulator of the controls.

But I think I may understand where the CFI is coming from. Until a recent change, 61.51(e)(1)(i)'s sole manipulator clause referred to "rated or has privileges." A number of folks (not the FAA) took the position that the 61.31(d) solo endorsement was a "privilege" that counted. I'm not aware of a Chief Counsel interpretation on it but the FAA ended up amending 61.51(e)(1)(i) to make it clear that it was referring to "rated or has sport pilot privileges" which, AFAIK, was always the intent.

Thanks everyone for your resonses, they were very helpful! I hoped I could log that time as PIC, which would help toward 15 hrs of PIC time required for MEI rating. But I had a feeling that I couldn't, so I asked here. Now, what do I do with that time in my logbook, cross it out or just leave it as is?

Depends on how you want to log it. Some prefer different methods.

If you want to log it as PDPIC, then I suggest making a new column and making a new entry today. Put your total PIC mistake in parentheses to signal a subtraction, then add that time into your new PDPIC column. Put in the remarks section the reasoning and dates effected.
or..

You can do the strike-through method and adjust your totals beneath. Both are accepted as reasonable entries. Don't use whiteout it usually looks unprofessional, or you are trying to hide something.

Explain in future interviews that your instructor decided to log it this way. As a "Student pilot" you did not know better. But now that you have evolved into a "professional mature" pilot you have corrected the time accordingly.

I did my commercial initial in a multi and I had to do the supervised PIC thing too. My CFI wrote it into my logbook under both the "PIC" and "Dual Received" columns. I now understand that to be incorrect, but the two different DPE's I used for my multi initial and single addon didn't say a word about it.

I don't think it supposed to count, but it depends on the FSDO. I know at least one that will count it towards the 15 hours of PIC (or at least as of two years ago allowed it).