Many people define these two terms differently though some think they are the same thing. At one point, I saw EMR defined as a digital copy (scanned copy) of a patient's paper records. This data was very cumbersome and hard to read or search. An EHR on the other hand is a database that encompasses one's health data that is captured into fields. See https://www.healthit.gov/faq/what-are-differences-between-electronic-medical-records-electronic-health-records-and-personal for these definitions. I don't think one is more comprehensive BUT an EHR is much more user friendly as it is searchable and can be used for evidence-based medicine and any other statistical analysis.
Hi, I have no comments to the EHR/EMR question, but related to the second part of your question; Health Information System(s) (HIS) is generally used more broadly than systems dealing with patient data, such as EHR/EMR. It could be anything related to information systems within the health domain. Common groups of systems include Health Management Information Systems (HMIS), EMR-like systems, various registries, systems for survey data, etc. Broadly understood, it can also include at least connections to lab, radiology, logistics, HR systems. A good starting point could be the HMN Framework and standards for country Health Information Systems: https://gateway.euro.who.int/en/the-health-metrics-network-hmn-framework-and-standards-for-country-health-information-systems/
So I would say HIS is a broad collection of EHR/EMR, HMIS, lab systems, radiology systems, etc etc.