What does it mean when we say that "a neuron connects to a certain cortical layer"? For example, if we say "a neuron connects to Layer V of the cortex", does it mean it synapses inside layer V, or does it imply that the soma of the neuron it synapses to is in Layer V (and it could synapse inside Layer II for example)?
Short answer
In complicated matters like neural connections in a layered cortex, it is all about semantics. One should carefully place statements in their context and deduce their meaning. 

Background
In Fig. 1 a schematic of the layering of the cortex is shown. The layers can be separated based on their histological and neurophysiological features:


Layer I: the molecular layer, contains primarily axons running parallel (horizontal) to the cortical surface. 
Layer II, the external granular layer, is composed of a mixture of small granule cells and pyramidal cells. The apical dendrites of the pyramidal cells extend into layer I and their axons descend into and through the deeper cortical layers.
Layer III, the external pyramidal layer, contains primarily pyramidal cells. Their apical dendrites ascend into layer I and their axons descend into and through the deeper layers.
Layer IV, the internal granular layer, consists almost exclusively of stellate neurons. It can be divided into subdivisions in many neocortical areas. Layer IV is the primary target for ascending sensory information from the thalamus.
Layer V, the internal pyramidal layer, consists predominantly of pyramidal cells. Apical dendrites may extend upward one or two layers, and the larger pyramidal cells may have dendrites extending into layer I. 
Layer VI, the multiform layer, contains an assortment of neuron types. The dendrites of pyramidal cells may extend into layer I. (source: Clinical Gate)


As the text in bold above shows, the apical dendrites of pyramidal cells can extend many layers. Hence, it is a matter of reading the text carefully to know what they are talking about.

In your example...


  [A] neuron connects to Layer V of the cortex...


...it would depend on the context, but in this isolated instance it simply reads that a neuron's axon terminates in layer V. Especially in histological texts a line like this would most likely have identified the axon and the axon terminal and have identified the layer where it terminates.

However, as said above, dendrites of pyramids may extend all the way up to layer I, so if a text would say...


  [A] neuron synapses onto a layer V neuron...


it is left undetermined exactly where that connection is made (can be layers I, II, III, IV, or V in the case of pyramidal cells).

What can be of help is the fact that when a text talks about... 


  ...a layer V neuron


...you can safely assume that its cell body is located in layer V, as a cell's location is determined by its cell body. Its dendrites can be in any layer (especially of the larger pyramid cells) and its axon may traverse all layers.   


Fig. 1. Layers in the cortex. source: Clinical Gate