OK - I'll try and provide some sort of answer to the several issues here. Whilst I suspect that it's really one for discussion on the talk-gb mailing list, I'll try and provide some answers to what I suspect are the key questions (and the answers to [your previous question][1] are also worth reading).
**Question: Is it right or wrong to add "`access=private`" to a "`highway=footway`" that is a "`designation=public_footpath`" just because it runs across private land?**
I'd argue that it would be irrelevant (and daft) to include a general access tag on a `highway=footway` unless there's some expectation locally that access other than foot might be allowed. Specifically, as you state, if something's a designated public footpath(4) then the landowner can't legally prevent foot access whenever they feel like it, so it's always going to be "foot=yes" (you can add that tag explicitly if you want - I often do - but actually it's implied by designation tag). The "runs across private land" thing is a complete red herring - you have legal access on foot.
**Question: Is it right or wrong to add "`access=private`" to a "`highway=track`" that is a "`designation=public_footpath`"?**
Here's where it gets interesting. If I see a "`highway=track`" in OSM in England and Wales with no access or designation tags at all I would not assume that public access (via any transport mode) was allowed - I'd put it into the "needs survey" bucket. It's highly likely (given the default of access in England and Wales) that there is no public access, but it's not certain either way.
If I see "`designation=public_footpath`" but no access tags; then like the footway above I know that there's an implied "`foot=yes`" so I do know that I have access on foot, but I don't know anything about other modes(1).
However, it's quite common for designated public footpaths to run along private farm tracks. Often there'll be a "private" sign to accompany the "public footpath" one. Here I'd suggest that an "`access=private`" tag is definitely justified (alongside "`designation=public_footpath`" and for clarification "`foot=yes`") to say that no, most transport modes aren't allowed but yes, foot access most definitely is.
**Question: What about Public Bridleways?**
They're similar to footpaths, except that the "`designation=public_bridleway`" implies "`foot=yes`", "`horse=yes`" and "`bicycle=yes`" (regardless of the rules for other traffic - there may still be a big "private" sign to keep out motor traffic). The "bicycle" part in no way indicates that it's passable by a cyclist on a normal road bike; just that they have a legal right to be there.
**Question: ... and Restricted Byways?**
Essentially as "public bridleways" but also allowing non-motorised vehicular traffic, so if you fancy a trip out in the coach and four, you're legally allowed to do so down a "`designation=restricted_bridleway`". Personally I'd also add the implied keys "`foot=yes`, `bicycle=yes`, `horse=yes`, `vehicle=yes` and (usually) `motor_vehicle=no`". The "`motor_vehicle=no`" part is usually needed in the parts of England that I tend to map because there's usually a sign excluding motor vehicle access to accompany the "restricted byway" sign (usually things are changed to "restricted byway" in order to explicitly exclude motor vehicle access). There is of course no guarantee that any particular non-motorised vehicle whill physically be able to get down a restricted byway.
**Question: ... and a Byway Open to All Traffic?**
The clue's in the name - this implies "`motor_vehicle=yes`" (i.e. you have a legal right to drive down there(2), along with other modes). Something signed as a "public byway" or just a "byway" is usually(3) a "Byway Open To All Traffic" ("BOAT").
**Question: What does `access=yes` mean?**
It means "whatever the sort of traffic that would normally allowed down here are". On a `highway=footway` it implies "foot=yes`" "`foot=yes`" not "`motor_vehicle=yes`". Your main dispute seems to be whether "`access=private`; `foot=yes`" or "`access=yes`" is correct on a "`highway=footway`" where foot access (and no other) is allowed. I'd argue that they both mean exactly the same thing ("`foot=yes`") and therefore you're both right. Personally, I'd just tag "`foot=yes`" here rather than "`access=anything`" - it'll cause less confusion. I'd also argue that it's also important to record the designation if it's a public footpath.
**Question: What does `access:foot=yes` mean?**
It probably means that the previous mapper was confused. I've always interpreted it as "`foot=yes`", but it's used by very few people in very few places.
**Question: What about footpaths, bridleways and tracks that aren't designated as public footpaths, bridleways or byways, etc.?**
Use access tags as best you can to describe the access rules that you know to be in force, based on your knowledge of the area. This might mean any one of a myriad of possible combinations - try and pick the best one based on what you see.
**Question: Do any of the 5 map styles available on OpenStreetMap.org show England and Wales style footpath designations and access rights?**
No.
**Question: Would it make any sense of one of them did?**
As things stand, not really - the osm.org styles all have to make sense internationally, and an English and Welsh public footpath rendering wouldn't even make sense across the whole of the UK.
**Question: Would it be nice if someone ponied up the cash for some web hosting to host a tileserver that did show this?**
Yes - feel free to contribute and organise. It'd have to be as a "local add-on" though.
**Question: Do any "off the shelf" routers understand England and Wales style footpath designations?**
Not that I'm aware of, though sanity in access tagging (saying "foot=yes" on footpaths and public tracks, omitting "access=private" when it is not needed etc.) will help greatly.
**Question: How do I make my Garmin car satnav avoid dirt tracks that I don't want it to route down?**
I find that removing them from the data before running mkgmap works for me. It does mean that you can't then use it for foot routing, but that's usually not an issue with a car satnav.
**Question: I have a disagreement with another mapper. What's the best way to resolve it?**
I'd start by trying to discuss it with them, and trying to understand each other's point of view. If that doesn't work, try the local mailing list (or forum, or facebook page, or whatever else your locality uses). If that doesn't work, try the OSMF [Data Working Group][2](5). It's unlikely that the DWG will be able to wave a magic wand, but they'll at least be able to offer a view independent of the disagreeing parties.
**Notes**
Firstly - all of the above are my views, based on a fair bit of mapping of public rights of way in the last 7 years or so. There *are* different approaches taken (see for example SK53's answer to your previous question - a quick chat on talk-gb or #osm-gb will reveal that there are no certainties here). If there are takeaway messages from any of this it's "don't assume there are any certainties; alway do an on-the-ground survey of rights of way; always talk to other mappers to try and appreciate their point of view".
(1) bicycle is an interesting one here - there *are* people who'll argue that you have a right to cycle on public footpaths, but let's leave that to one side for now.
(2) in the absence of anything else preventing it, such as a traffic regulation order.
(3) I don't believe I can think of a single exception.
(4) in England and Wales - other countries have far more sane access rules.
(5) full disclosure - that currently includes me.
[1]: https://help.openstreetmap.org/questions/37777/rights-of-way-england-and-wales
[2]: http://www.osmfoundation.org/wiki/Data_Working_Group