The Deathwatch rulebook often introduces mechanics that actually don't apply to Space Marines, because a lot of it is copied from earlier 40kRPG rulebooks where the PCs are not Astartes and therefore the mechanics are relevant. For example the bracing mechanic is explained on p140 and p238, even though every Space Marine has Bulging Biceps and therefore never needs to brace before firing a heavy weapon.
Similarly the Fear paragraph is essentially copy-pasted from earlier 40kRPG rulebooks, giving a general introduction to the Fear mechanics and how they would usually apply to PCs (and NPCs). See this comment for a comparison of the Fear text across rulebooks. Since Space Marines know no fear, the And They Shall Know No Fear... section then introduces the rules specifically for Space Marines.
Note from the Fear paragraph that the triggering event for taking a Fear test is "when a PC is confronted by such a frightening event or adversary". Note also that the wording in the And They Shall Know No Fear section specifically says "when facing a Fear-causing foe" and "when facing a Horde of Fear-causing creatures". Therefore the special rules for Space Marines kick in at the point of facing a Fear-causing enemy, which is the same point the normal Fear rules trigger. So at this point a normal non-Astartes PC or NPC takes a Fear test. A Space Marine (following the special rules) immediately suffers -10 WP or 1 Cohesion damage per level of Fear, and otherwise continues as normal.
Note also that a normal PC also takes a Fear test when encountering a frightening event, some examples of which are given in the Very Bad Things section on p278 (e.g. viewing the scene of a gruesome murder is Fear 1, staring unprotected into the Warp is Fear 4). But per the And They Shall Know No Fear section, Space Marines specifically do not experience Fear effects from events, only from enemies - "For a Battle Brother, Fear has the following effects, and then only when in the presence of Fear-causing foes".
You said you play Fear the way Adeptus-B describes - but he agrees with me that RAW Space Marines do not take Fear tests. See this comment a couple before yours. He also says he house rules that they do take the test, and of course you are welcome to do so as well, but we are talking about RAW.