Clustering GPS data using DBSCAN but clusters are not meaningful (in terms of size)

I am working with GPS data (latitude, longitude). For density based clustering I have used DBSCAN in R.

Advantages of DBSCAN in my case:

1. I don't have to predefine numbers of clusters

2. I can calculate a distance matrix (using Haversine Distance Formula) and use that as input in dbscan

   library(fossil)
   dist<- earth.dist(df, dist=T) #df is dataset containing lat long values
   library(fpc)
   dens<-dbscan(dist,MinPts=25,eps=0.43,method="dist")
   

Now, when I look at the clusters, they are not meaningful. Some clusters have points which are more than 1km apart. I want dense clusters but not that big in size.

Different values of MinPts and eps are taken care of and I have also used k nearest neighbor distance graph to get an optimum value of eps for MinPts=25

What dbscan is doing is going to every point in my dataset and if point p has MinPts in its eps neighborhood it will make a cluster but at the same time it is also joining the clusters which are density reachable (which I guess are creating a problem for me).

It really is a big question, particularly "how to reduce size of a cluster without affecting its information too much", but I will write it down as the following points:

1. How to remove border points in a cluster? I know which points are in which cluster using dens$cluster, but how would I know if a particular point is core or border?
2. Is cluster 0 always noise?
3. I was under the impression that the size of a cluster would be comparable to eps. But that's not the case because density reachable clusters are combined together.
4. Is there any other clustering method which has the advantage of dbscan but can give me more meaningful clusters?

OPTICS is another alternative but will it solve my issue?

Note: By meaningful I want to say closer points should be in a cluster. But points which are 1km or more apart should not be in the same cluster.