"every two days" and "every second day"

You can use "every other", as given by Merriam-Webster, for example

I go to the club every other day.

and the "other" does not need to be stated.

Another phrasing is with "alternate", also given by Merriam-Webster, for example

I walk to work on alternate days.

Neither of your phrases describes case 3 because the boundaries between the pairs of days are artificial. If you do something on day 2 and also on day 3 then you have done it every day for days 2 and 3. If you then leave it until day 6 you have only done it once in days 3, 4 and 5 so you have not done it "every second day" or "every two days" except on average.

If you say you are doing something every two days you either have to do it on every odd numbered day or on every even numbered day, it doesn't matter which but you have to be consistent.

If you say "every second day" then it initially appears that you only do it on even numbered days; but when you start counting those days is arbitrary unless the start time is fixed. So if you say "We are going to clean the deep fat fryers every second day" and your restaurant is open 24/7 then which day you start is up to you. However if you are at a week long music festival starting on a Saturday and a given band is appearing on every second day you would expect them to appear on Sunday, Tuesday and Friday.