send R diagnostic messages to stdout instead stderr

Using sink. Here's a modification of your code:

sink(stdout(), type = "message") # sink messages to stdout
print("using print")
cat("using cat\n")
message("using message")
warning("using warning")
sink(NULL, type="message") # close the sink
warning("after ending sink") # this will be the only thing in your err file
q("no")

The OP showed the execution happening via the Rscript command and using some I/O redirection. If you want to use redirection to log everything and only show to console on error, the best method I've found is to use || to check if the script had non-zero exit status before printing to screen:

Rscript myrscript.R > temp.log 2>&1 || cat temp.log

This method relies strictly on the exit code for printing, which only partly gets around message() going to stderr, but I thought this example helpful to mention since messages won't necessarily trigger a non-zero exit status and you can continue to log quietly with this method.

If you'd like to go one step further and keep appending to a single log file, then this will work:

Rscript myrscript.R > temp.log 2>&1 || cat temp.log && cat temp.log >> persistent.log && rm temp.log

The pseudocode for this command is:

1. Redirect stderr and stdout into temp.log
2. If command had non-zero exit status then write to screen
3. Then redirect the contents of temp.log into your persistent.log
4. Then remove temp.log

While this is very likely not a best practice, you could override message with a version that writes to stdout() by default, right?

message <- function (..., domain = NULL, appendLF = TRUE) 
{
    args <- list(...)
    cond <- if (length(args) == 1L && inherits(args[[1L]], "condition")) {
        if (nargs() > 1L) 
            warning("additional arguments ignored in message()")
        args[[1L]]
    }
    else {
        msg <- .makeMessage(..., domain = domain, appendLF = appendLF)
        call <- sys.call()
        simpleMessage(msg, call)
    }
    defaultHandler <- function(c) {
        cat(conditionMessage(c), file = stdout(), sep = "")
    }
    withRestarts({
        signalCondition(cond)
        defaultHandler(cond)
    }, muffleMessage = function() NULL)
    invisible()
}