Confusion about "host bus controller" and "host bridge"
Keep in mind the flow of information must follow the ISO 7 layers.  You are mainly concerned here with the lowest level, i.e. the Physical Layer and that Protocols exist in each Layer, so all layers handle information but mainly Session and Physical layer for this context. 

To avoid complications here is another simple model.



Read an overview on the OSI 7 layer model to get a broader perspective for any computer.



Regarding PC's

Communication Layers and Flow.


  General comment: One cannot understand ANY meaning of "Bridge" unless you define the traffic and the Bus ( or physical carrier of data) on each side. 



Just as Host can meaning anything until you define its "context" or frame of reference. Again refer to OSI 7 layers.  


edit


A Host can be physical, logical, virtual, anything 
An Adapter  allows connection and communication from any target to another target. It must be specific and  understand the Physical requirements and the protocols of any/every layer relevant to and from the target device.
This regardless if it is physical or virtual but enabled by the Host with higher level permissions.  (e.g. USB-Serial, SCSI-PCI)
It may be integrated or broken down into "Functional blocks" 


then it becomes a "HA" and may include a bus "bridge" , hence the names PCI Host bridge and SCSI HBA
thus the flow and logical/physical/functional blocks depends greatly on specific hardware and the intent of the explanation with respect to understanding.


as a smart phone is quite different from a motherboard.




A PCI Root Bridge is represented in UEFI as a device handle that contains a Device Path Protocol instance and a PCI Root Bridge Protocol instance.

"host controller, host adapter, or host bus adapter (HBA)" are all the same except historically HA's are closer to the target device and HC's is integrated and closer to the CPU. But this gets into design evolution from cards to chips to super IO chips.

USB3 is an evolution where targets can initiate traffic and connect target to target with permission from the CPU/OS/"User" privies etc.