In terms of academic careers, does the department type matter between Computer Science, Engineering, or Information? One of the oddities of working in an interdisciplinary field is that PhD programs - and ultimately departments where you would like to work in the future - are often offered under different departments. For my personal example, my field is Human Computer Interaction (HCI), and I have the option to join a PhD program in (that is, have received admission offers from) departments that are housed within colleges of Computer Science, Engineering, or Information. "Rankings", to the extent that anyone cares about them, exist separately for each classification, and so they are all "top 5" within their own classification and have a good reputation in the field.

My question is: in considering a plan to seek future academic jobs after earning a PhD (especially as a professor at an R1), is the name of the home department/college that hosts their program something that one should weigh in their decision making? How might the type of department that one gets their PhD from effect one's future career?

Bonus: If department type impacts availability of internships (especially those valuable to an eventual faculty application), or just having the alternative to go into another area of industry/consulting, that could be important for a person to know and consider. "Plan A" is primarily positions in academia, so an answer need not address this to be a good or accepted one.

Note: The  focus of research, topic, equipment, and even coursework will be largely comparable, faculty working in every program have degrees in widely differing topics (including some in Psychology, Art/Design, and Business/Management), and current faculty even often publish in the same conferences.

For example, if you've been involved with a hiring committee in Computer Science, have they evaluated applicants differently if their degree was not specifically in "Computer Science" and instead in Information or Engineering? Or similarly, do Engineering hiring committees have a preference for people with degrees from other Engineering departments? Issues at the administrative culture level? Have you found that informal cliches or hierarchies form between groups?

Ultimately I'd like to know if this a distinction I need to seriously consider in picking a program, or if I can instead focus on the dozen-other factors one should consider to compare options instead.
My short answer is that if you're staying in academia, it probably doesn't matter much at all. If you're considering leaving, then CS > "Engineering" > IS (for many tech companies, at least), to a mild-to-moderate extent.

My longer answer is that it depends how you sell yourself. The material you learn will likely be very similar, but different communities will focus on different aspects of HCI. This will probably be most evident in the research coming out of each group. To that extent, I recommend taking a good look at the research coming out of each program. The focus of the researchers will give you a good idea about the differences between the programs.