What are the "standard unambiguous date" formats for string-to-date conversion in R?

This is documented behavior. From ?as.Date:

format: A character string. If not specified, it will try '"%Y-%m-%d"' then '"%Y/%m/%d"' on the first non-'NA' element, and give an error if neither works.

as.Date("01 Jan 2000") yields an error because the format isn't one of the two listed above. as.Date("01/01/2000") yields an incorrect answer because the date isn't in one of the two formats listed above.

I take "standard unambiguous" to mean "ISO-8601" (even though as.Date isn't that strict, as "%m/%d/%Y" isn't ISO-8601).

If you receive this error, the solution is to specify the format your date (or datetimes) are in, using the formats described in ?strptime. Be sure to use particular care if your data contain day/month names and/or abbreviations, as the conversion will depend on your locale (see the examples in ?strptime and read ?LC_TIME).

In other words, is there a better solution than needing to specify the format?

Yes, there is now (ie in late 2016), thanks to anytime::anydate from the anytime package.

See the following for some examples from above:

R> anydate(c("01 Jan 2000", "01/01/2000", "2015/10/10"))
[1] "2000-01-01" "2000-01-01" "2015-10-10"
R> 

As you said, these are in fact unambiguous and should just work. And via anydate() they do. Without a format.

As a complement to @JoshuaUlrich answer, here is the definition of function as.Date.character:

as.Date.character
function (x, format = "", ...) 
{
    charToDate <- function(x) {
        xx <- x[1L]
        if (is.na(xx)) {
            j <- 1L
            while (is.na(xx) && (j <- j + 1L) <= length(x)) xx <- x[j]
            if (is.na(xx)) 
                f <- "%Y-%m-%d"
        }
        if (is.na(xx) || !is.na(strptime(xx, f <- "%Y-%m-%d", 
            tz = "GMT")) || !is.na(strptime(xx, f <- "%Y/%m/%d", 
            tz = "GMT"))) 
            return(strptime(x, f))
        stop("character string is not in a standard unambiguous format")
    }
    res <- if (missing(format)) 
        charToDate(x)
    else strptime(x, format, tz = "GMT")
    as.Date(res)
}
<bytecode: 0x265b0ec>
<environment: namespace:base>

So basically if both strptime(x, format="%Y-%m-%d") and strptime(x, format="%Y/%m/%d") throws an NA it is considered ambiguous and if not unambiguous.