In terms of specificity, while it is debatable, I tend to agree with the others that the net is more specific in this instance.
Familiars typically do not have DCs associated with their attacks in 5.5. DCs in terms of statblocks are usually reserved for creature-specific actions that are not attacks.
The net on the other hand is an attack (in that you replace the attack action to use it). A net in this fashion applies a DC to an attack and that is specific and generally an anomaly in 5.5E with the other exception being an unarmed strike, which is also a very specific attack.
I also think when the Invocation states "if the familiar forces a creature to make a saving throw" it means specifically things the familiar can force - Scare on a Quasit, Sting on a Pseudragon, Heartsight on a sprite. I don't think it means any general save as a result of the environment that are not "forced" by an ability of the familiar.
Further if we are to take this to its limit then a familiar that made an unarmed strike would use the Warlock spell save DC for a push or shove, or a familiar that used a wand or threw caltrops would use the Warlock spell save DC. If the familiar pushed an enemy into your Cleric's Spirit Guardians it would use the Warlock DC instead of the Cleric DC. Likewise if a familiar used deception to cause an enemy to walk into weak floor plank which collapsed you would use the warlocks DC instead of the DC for the weakened floor. I don't think these things constitute a familiar "forcing" a save, I think they are the object or other environmental effect forcing the save.
sorry - i do obviously disagree with you - but also there are a lot of inconsistencies with what you are arguing:
1. i may need some clarification on what "familiars typically do"–find familiar is a specific rule that creates a 0 CR beast that cannot attack (specifically note, "cannot attack" and not "cannot take the attack action" which are two different things). example every single familiar (which is still a CR 0 beast) can grapple (pending size of the target), which would be a general rule: grappling.
2. the net on the other hand is NOT an attack. it has a specific rule that you "can" use it as a part of the attack action, but it most certainly is not an attack (or a weapon for that matter), by RAW. furthermore, an attack requires an attack roll (not a saving throw) and no where in the net description do you make an attack roll. it is not an attack and just because you use something through an attack action does not make it an attack.
3. the invocation does specifically state "If the familiar forces a creature to make a saving throw, it uses your spell save DC" and nowhere in that sentence does it specifically say "except when using and item" or "except when grappling" or "except when doing X"–nor does it say "If the familiar forces a creature to make a saving throw using one of its actions"–i am not sure what you are emphasizing with force, if the familiar's action requires that another creature make a saving throw, how is it not forced?
4. a lot here, will try to tease out:
4a. yes, use warlock save DC to shove a creature up to 1 size category larger than the familiar, there is no mechanic for push, not sure what that is.
4b. yes, use warlock save DC for caltrops (exactly the same as a net)... although this one could be argued as the caltrops stand alone after use. the familiar is not force a save, a creature would have to walk into the area. i suppose this one is an actual grey area, and a fair point.
4c. spirit guardians specifically uses the cleric DC and is not an effect forced by the familiar (is this up for debate?)
4d. using an opposed skill check is not forcing a saving throw (is this up for debate?)