Primer is to be applied to bare wood to seal it before any other paint types.
So for your pine door :-
Treat any knots with Knotting (stops the sap leaking out)
Prime
Undercoat (may need more than one,depending on colour)
Topcoat
And sand down between coats (when paint is dry) for a smooth finish - just rub lightly with fine paper,as it is almost impossible to paint without having dust land on the wet paint!
Then main difference is primer contains more oil and is generally thinner...its designed in most cases to soak into the surface
Undercoat carries more pigment and its main roles are obliterating the previous surface colour and to 'key' onto the existing surface, providing a suitable base for the topcoat.
Deepends on what topcoat you intend to use...if its gloss id say yes...you need to undercoat, this will help remove surface texture it will also obliterate the surface....using two coats of gloss may not totally cover.
If your going to use satin or eggshell finish then two top coats should be ok as it has more 'body' than gloss.
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