In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold.

Could you please point me to texts doing the same kind of construction (not necessarily using densities) in vector bundles over

arbitrary(i.e. not necessarily compact anymore) Riemannian manifolds for the connection Laplacian?

I am mostly interested in the clear definition of the object and the statement of its basic properties. Do the positivity and minimality properties of the heat kernel on functions have analogous properties in bundles?

localsince Laplacian is defined only locally. Thus the construction you learned can be extended to the interior of any manifolds. For the construction of heat kernel over manifold with boundary, there are many references available. One thing you may clarify is that you probably need some boundary condition imposed on the metric. Otherwise the Laplacian on the buondary is technically no different from the Laplacian on a different submanifold. – Bombyx mori Sep 13 at 21:23