The comment from @gung is directly on point. Your two methods are measuring different things.
Method 1 is, as noted in the answer from @AlexandreCazenave-Lacroutz, "the classical way to make inference." But think closely about what Method 1 is testing. It's whether your variable of interest is "significantly" related to heart rate in a very specific sense: if there were really no relationship and you repeated the experiment (with the same number of cases) multiple times, you would find such a large relationship by chance in less than 1 out of 20 experiments (at p < 0.05). With small sample sizes and particularly with correlations among predictor variables, it's quite possible for Method 1 to miss a true relationship between your variable and heart rate; a larger sample size might let you document its "significance."
Method 2 has to do with predictive performance. It's quite possible--even likely--that a variable that doesn't pass the strict test of Method 1 could still improve performance when added to other predictor variables. For predictive modeling it's generally considered good practice to include all variables that are reasonably related to outcome provided that you are not overfitting. For example, Harrell in Regression Modeling Strategies lists as one recommended approach in this context (page 89):
fitting fully pre-specified models without deletion of “insignificant” predictors.
In summary, Method 2 can document whether your variable bears some relationship to heart rate, in that including information about it provides a better estimate of heart rate. Method 1 documents whether the combination of the magnitude of that relationship with the size of your data set is sufficient to rule out a chance relationship at a specified level of significance. Which is "more appropriate" depends on which of those criteria is more important to you.
In response to comment:
You need to make sure that your model comparison test in Method 2 has sufficient sensitivity to see differences. For example, Harrell explains (see the links from that page for more details):
The two most commonly used resampling methods are cross-validation and bootstrapping. To be as good as the bootstrap, about 100 repeats of 10-fold cross-validation are required.
Note that bootstrapping (sampling with replacement) is more closely akin to re-sampling from the original population than is cross-validation (a form of sub-sampling without replacement). See this page for some justification. I typically use cross-validation for things like choosing penalty values in ridge regression or LASSO, then use multiple bootstrap samples to repeat the model-building process, with performance testing of the multiple models on the original data set. That's a good way to estimate bias, "optimism" in estimates due to overfitting, and how well your modeling process would be likely to work on a new sample from the original population. See this page for an outline of this approach and links to further documentation.