Why do we refer to our estimates in terms of precision? Open any statistics textbook and it will urge the need to check the 'precision of our estimates'.
Take the following random variable:
set.seed(55)
x <- rnorm(n=10, mean=0.3453, sd=1)

x
 [1]  0.4654391 -1.4670769  0.4968830 -0.7739210  0.3472082  1.5338185 -0.1600439  0.2460656  0.6506532
[10]  0.5437097

mean(x)
[1] 0.1882736

t.test(x ,alternative="two.sided",mu = 0,conf.level=0.95)$conf.int
[1] -0.4025211  0.7790682

I would describe the mean of this variable as being very precise, in fact precise to 7 decimal places. I would not describe it as being very accurate, as its 95% confidence interval spans a large range of the data. However, in statistics-speak, the mean of the variable would be described as being imprecise.
So why don't we refer to the 'accuracy of our estimates', instead of the 'precision of our estimates'?
 A: I'm not a statistician. Here's a stab anyways.
Basically, we generally don't know the true value of some parameter. So we can't talk about how close we are to the truth (i.e. we don't know how accurate our measure is). Instead, we collect a bunch of data and make estimates of the truth. If those estimates vary a lot, then that's the definition of imprecision. If they vary little, then we have a precise estimate. But is it accurate? Who knows? We can only hope that we chose the best methods that lead us to an accurate assessment of the truth. If we have, then we presume we have an accurate measure, but we're only sure about how precise it is.
To clarify, certainly statisticians' idea of precision has nothing to do with how many digits we calculate a mean to. Instead, it's roughly how likely we would be to get a similar estimate if we resampled.
A: It is obviously not correct to say that the mean is imprecise. You are correct in concluding that R gives a very precise measurement of the mean.
It is valid to question the use of the word precision for estimates. 
Though your example misses the point in that case.
"The mean is precise" and "The mean gives a precise estimate" are not identical.
The former is what you use in your example and your reasoning is correct. However that is not what is beeing said in statistics.
What is meant by "precision of our estimate" is that the mean is not an estimate rather it is an estimator, obviously.
Therefore an accurate estimator can give us a precise estimate of the a real parameter.
Now the question becomes, can the estimate be precise (in the meaning that it is close to the real parameter?) if the precision of the estimator (mean) has nothing to do with the closeness to the unknown parameter?
I personally think so and looking at definitions of "precise" do support me in this.
Hence I think that an estimator can be accurate or not. It can also be precise if the calculation is (for example the mean of this sample).
An accurate estimator, such as the mean, can give an estimate. This estimate can be precise if the estimator is accurate in estimating an unknown parameter.
A: Your instinct that something is wrong with the common parlance is on point. Precision is not a property of an estimate. Precision is a parameter, meaning it exists in the long-run. An estimate exists as a single quantity computed from a single sample, so it cannot vary. In practice, when we say that an estimate is precise, we mean that it has been drawn from a population distribution with a relatively small standard deviation (aka, standard error). We draw that conclusion by estimating the population variance from the sample. So the language is quite imperfect.
What's more, the distribution we're describing isn't even the distribution of the parameter of interest$-$the sampling distribution is a distribution of estimates, from which the current estimate has been drawn. The variance of that distribution is a combination of the variability in the population, the size of the sample, the efficiency of the estimator, and the reliability of the measure (assuming you use the same measure over and over "in the long run").
The accuracy of the estimate does exist: it's the estimate minus the true value. Unlike the standard error, it has both magnitude and sign (i.e. can be negative). Also unlike $SE$, it is not a parameter and cannot be estimated. (With the caveat that, given multiple estimates of the same parameter, as in a meta-analysis, you can pool information, cancel out some of the error, get a more precise estimate, and then subtract that from any one estimate to "estimate" its accuracy.) To estimate something, you need Fisher information; Fisher information is parameter information; parameters exist in the long-run; accuracy is a function of a parameter but exists only for the particular estimate in hand.
So, estimation theory gives no means of determining accuracy in practice, short of a population census or a made-up example. And that's why statisticians don't refer to estimate accuracy.
