I would say that all four of the sentences in the question are valid, and the two in "Example 2" carry much the same meanings as the corresponding two in "Example 1". I would not agree that the forms using "will" indicate a greater degree of certainty, assurance, or probability than the forms where "will" is omitted. I will say that I prefer the forms that use "will", and think they are better writing in any circumstance I can think of.
I neither know nor care which of this fits the "first conditional". The use of the "numbered" conditional forms neither captures all the valid forms, nor do I find it generally helpful. Particularly when a learner assumes that the numbered forms are the only valid choices, the device is harmful. In any case learners should understand that no native speaker ever learns conditionals in this way, and no fluent speaker is limited to the numbered forms.
See https://ell.stackexchange.com/a/334537/91457
Both the forms
- If {X} you {verb} Y
- If {X} you will {verb} Y
express a general rule, or a confident prediction (either form may be used for either case). Both are dealing with possible future action.