what is the difference between euclidean distance and RMSE?
Let's say that RMSE and euclidean distance are not of the same type, even though their formulations are close. RMSE is a loss function, while euclidean distance is a metric. See this question on Cros Validated to better understand the difference between a loss function and a metric: a loss function is generally based on a reference metric.

Euclidean distance is a metric, so it quantifies the distance between two observations.
RMSE is, as the name suggests, the root of the mean of the squared error between a true value and a predicted value, over a range of observations. RMSE is generally intended for model performance assessment.

Your confusion probably comes from the fact that, if you apply the RMSE formula to your set of coordinates (longitude and latitude), it gives the euclidean distance. But this would not be called RMSE in that case, rather euclidean distance.

In your case, you may use the RMSE of euclidean distance as loss function. The error made by your predictor is the euclidean distance, and your loss function would be the RMSE of these errors.

Defining a loss function is strongly problem-specific. First, you need to determine which metrics to use as error function. In your case, the euclidean distance between the actual position and the predicted one is an obvious metric, but it is not the only possible one. For instance, you could use the squared or cubed euclidean distance in order to give more weight to cases that are not well predicted. You could also design an ad-hoc metric to consider:


assymmetry, e.g. to be more tolerant on errors towards the east side than the west side
anisotropy, e.g. to be more tolerant on errors in the latitude axis than the longitude
uncertainty tolerance, e.g. error is null if the euclidean distance is less than a tolerance, and is equal to the euclidean distance otherwise
etc. (depending on your problem)


The subsequent metric will allow you to evaluate the performance of the trained model over a test set, and will therefore give you a disitrbution of errors. You need to shrink this distribution to a single scalar value used as global model performance: you can pretty much choose any scalar that represents something about the distribution:


Mean squared error (MSE) or its root (RMSE) - MSE is faster computed but RMSE has the advantage or having the same dimension as the error function (a distance in your case)
Other types of $\mathcal{L}_p$ norms (RMSE is the $\mathcal{L}_2$ norm)
Any quantile of the distribution (median, 75%, 95%, maximum value)
Whatever you think is relevant!