The proper way is the one described in the recipe you are making.
All of the variations you listed above will give you a fruit puree, for any berry (or even other fruit) you choose. The purees will of taste differently depending on how you make them, but none of them is somehow less "proper" than the other.
It is the recipe author's job to choose a preparation method in which the puree's texture and taste harmonizes best with the rest of the recipe. So, just follow it, and you will get the intended result.
There are two cases in what you listed in which the combination of preparation steps could be problematic (so if you see it in a recipe, you should choose another recipe). First, if you have small seeds (such as a raspberry or blackberry) and need a seedless puree, but use a bladed implement like a food processor or a blender before straining. In that case, you will most likely end up with sharp seed pieces that are left in after straining, defeating the purpose of a seedless puree. Second, thickeners need the correct conditions to work. As you listed pectin, you have to make sure that you are within the correct ratios of sugar and acidity for the type of pectin you are using, and you have to warm the puree to the needed temperature. With other thickeners, you have to again ensure that their requirements are met.