"Bootstrap validation"/"resampling cross-validation" is new to me, but was discussed by the answer to this question. I gather it involves 2 types of data: the real data and simulated data, where a given set of simulated data is generated from the real data by resampling-with-replacement until the simulated data has the same size as the real data. I can think of two approaches to using such data types: (1) fit the model once, evaluate it many times on many simulated data sets; (2) fit the model many times using each of many simulated data sets, each time evaluate it against the real data. Which (if either) is best?
Short answer: Both validation techniques involve training and testing a number of models.
Long answer about how to do it best: That of course depends. But here a some thoughts that I use to guide my decisions about resampling validation. I'm chemometrician, so these strategies and also the terms are more or less closely related to analytical-chemical problems.
To explain my thoughts a bit, I think of validation as measuring model quality, and of training as measuring model parameters - this leads to quite powerful analogy to every other kind of measurement.
There are two different points of view to these approaches with respect to validation:
If you take that thought further, your approach (1) tells something about how much predictions of the same model vary for different samples of size $n$.
Your approach (2) is closer to the usual approaches. But as Momo already wrote, validation usually wants to measure the performance for unknown cases. Thus you need to take care the testing is not done with cases that are already known to the model. In other words, only the left-out cases are tested. That is repeated many times (each model leaves out a different set of cases) in order to (a) measure and (b) average out as good as possible the variations due the finite (small) sample sizes (for both testing and training).
Not that resampling validation allows you to measure performance for unknown samples. If in addition you want to measure the performance for unknown future samples (instrumental drift!), then you need a test set that is measured "in the future" i.e. a certain time after all training samples were measured. In analytical chemistry, this is needed e.g. if you want to find out how often you need to redo the calibration of your instrument (for each determination, daily, weekly, monthly, ...)
Bootstrap vs. cross validation terminology:
Both can have some kind of stratification. Historically, the splitting for cross validation (at least in chemometrics) has often been done in a non-random fashion, e.g. a 3-fold cross validation of the form abcabc..abc (data set sorted wrt. the outcome) for calibration/regression if you have very few cases (physical samples), and you want to make sure that your whole data range is covered.
Both techniques are usually repeated/iterated a number of times. Again for historical reasons and at least in chemometrics, k-fold cross validation often means training and testing k models (each tested with the 1/kth of the data that was not involved in training). If such a random splitting is repeated, people call it iterated or repeated cross validation.
Also, the number of unique samples can (approximately) be chosen: for cross-validation via the $k$ of $k$-fold or the $n$ of leave-$n$-out cross validation. For bootstrap, you can draw more or less than $n$ samples into the subsample (this is rarely done).
Bootstrap resampling is said to be better (faster convergence, less iterations needed) than iterated $k$-fold cross validation. In a study for the kind of data I deal with, however, we found little overall difference: out-of-bootstrap had less variance but more bias than iterated $k$-fold cross validation.
I don't know about "best" (which probably depends on what you use it for), but I use bootstrap validation to estimate error on new data the following way ( third way if you like):
What is out of bag is not always clearly defined. Often it is all those observations that weren't part of the training set. More stricter would it be (I use it this way) to only have observations in the oob sample that have a realisation of the whole predictor vector that is not part of the training set (which is especially useful if you have many factors). Even stricter is to use an oob sample that contains only those observations who have a different realisation of the predictor variable on the predictors chosen in the model (especially useful if the model is found with some variable selection procedure, e.g. trees).
Then I usually repeat this a number k of times and aggregate results over the k-folds (mean or median or whatever statistic is handy). The model chosen this way can then be fitted to the overall data set (as in your option 2) to additionally gauge if there still is a tendency to overfit (the performance measure should be not too far off from the bootstrap samples).
If I have more models or a parameter grid or similar, I fit them all to each training set and evaluate them all on each oob sample. It is also possible to not use a training set twice, but for every model or tuning parameter combination to draw a new training/oob pair.