14 cfr 61.51
(g)Logging instrument time.
(1) A person may log instrument time only for that flight time when the person operates the aircraft solely by reference to instruments under actual or simulated instrument flight conditions.
(2) An authorized instructor may log instrument time when conducting instrument flight instruction in actual instrument flight conditions.
(3) For the purposes of logging instrument time to meet the recent instrument experience requirements of § 61.57(c) of this part, the following information must be recorded in the person's logbook -
(i) The location and type of each instrument approach accomplished; and
(ii) The name of the safety pilot, if required.
(4) A person can use time in a flight simulator, flight training device, or aviation training device for acquiring instrument aeronautical experience for a pilot certificate, rating, or instrument recency experience, provided an authorized instructor is present to observe that time and signs the person's logbook or training record to verify the time and the content of the training session.
So back to when can the approach be logged, the way I read it, 61.51 (g)(1) would be actual instrument if the flight at that point was conducted solely by reference to instruments. It could be 10 miles in haze or 1 mile in haze. If your primary reference is the instruments, then it is instrument time.
Keep in mind that instrument flight can be conducted even though the weather conditions are better than VFR minima. For example, at night with no ground reference, over water, in haze, in snow or precip.
So my short answer, is that the actual weather at any point is secondary, and the primary criteria is whether you are operating solely by reference to instruments. And I would add that if you are in and out of clouds or precip, or smoke or whatever, you need to fly by instruments because your outside reference is unreliable. You can still scan outside the plane, even though you rely on the instruments.
EDIT to address comment by original poster:
What weather does the FAA require (assuming that I'm not wearing a view limiting device) in order to log the approach for currency requirements? The issue is not the weather but how you are conducting the flight, and when you fly solely by reference to instruments, you may log it as instrument time.
For instance, if I am cleared for an ILS in visual conditions, can I log it?
It depends upon what reference you are using to fly. If solely by instruments, then you can log it.
What if I start the approach in the clouds and break out at 1,500 ft and continue the approach? 1,000 ft? Again, it depends upon what your reference for flight is. If instruments, then log the time. If you break out at 1000 ft, you might still be in non-VFR conditions, depending upon the airspace, etc.
Before/after the outer marker?
What is your flight reference?
150 feet above minimums?
What is your flight reference?
I think that you get the idea....
Not really. The regs are reasonably clear in that they say you can log when you fly by instrument reference. Just because you head pops out of a cloud does not mean you are not flying by instrument reference.
To editorialize, I would be careful looking at FAA Letters of Interpretation. A local pilot did exactly that, and had his ATP and CFI revoked. Look at the regs. Because the regs, not some lawyers opinion on a given day, is what the NTSB will be applying at your hearing.