The verb おく means "leave an interval (in space or time)".
So, how you interpret the "interval" is the key. In usual settings, as you said, we take 一日おき as "every other day" or "two days later" because we treat a "day" as a unit (a) and leave one day between. But there is technically another way of interpretation, that if you focus on the time points two events take place, you could also say 一日おき when they're apart by one day duration (b). In this case, the interval is only one day (24h).
You can actually pick a usage of (b) from a random Google search result, as in:
According to the NHK放送文化研究所's article, the percentage those interpret 一日おき as "every day" is gradually increasing as generations go down, though the great majority still think it "every other day". It also points out an interesting fact:
Meanwhile, when we conducted a survey about the phrase 特急は１時間おきに出発します, the interpretation "every hour" was predominant in general (across generations)...
Um, forgot to write the conclusion... Ultimately, I have to say that whether your その夕食から1日おいて is one or two days later depends on context.
Some people think I don't answer the question correctly. Well, just to clarify, I myself believe その夕食から1日おいて should always mean "two days later", but according to a 2007 survey, we should prepare to face avg. 10% chance to see it in "one day later" meaning.