Bit Protector Variations
The HMS' Bit Protector's (BP) role is binary. Firstly, it fastens the Running Core to the Attack Ring, sandwiching the Weight Disk between the two and keeping the whole thing together. Secondly, it is what attaches to the HMS Launcher and gives it a close grip so the Beyblade doesn't immediately fall off. BP categorization below is practically identical between both Hasbro and Takara releases. Older releases were generally smaller, newer releases were larger. The only difference among the two brands is that Hasbro's early releases had a "B#" type stamp while Takara was "A#" almost exclusively throughout.
Honourable Mention and Credit: I was at a loss early on, BP compatibility seemed more or less random, and while it still needs to be taken case-by-case, it was only with @4DFury's insight that we came to learn that molds had nothing to do with the ID at the back or specific Beyblades, but everything to do with diameter of a given BP and their era of release.
Note: It is highly recommended that you use the tightest fitting BPs for your HMS combos as that will greatly reduce the chance for RC prong breakages in battle. BP and RC configurations are best taken case-by-case. In the end, BP Type only helps understand tightness/looseness generally, and specific BP or RCs wear or overall chemistry can vary. Some parts simply work together well, and it can't be quantified succinctly by diameter alone.
First BP Type - Small Mold
These set of BP molds start with A-123 Gaia Dragoon MS, the first HMS release, all the way up to the Gimmick Series. The latest small in my possession is MA-14 Advance Eterner. As the name implies, these are the smallest in diameter. This extends to Hasbro products as well.
These are often the tightest fitting. Regardless of RC mold, these BP will fit very snug with little-to-no rattling. This is the ideal scenario to reduce the chance of breakage of either the BP or RC prongs.
In some cases this tightness is actually too much, and you won't be able to properly twist the BP into the RC prongs at all. In those cases it is absolutely recommended to not force it and find a different BP that does not require very much force at all. Being too tight can also create a point of breakage.
These BP do have the potential to provide the most close-fit, but be cautious to not use ones that are too tight and only twist in with a lot of force.
Diameter of "Small Type"/"Mold 1": 8.4mm-8.5mm
Two Small Mold BP: Gaia Dragoon MS' and Advance Eterner's
Second BP Type - Large
These BPs have the widest diameter, significantly wider than Small. These are typically present in the HMS Gimmick Series. This means anything from MA-15 Phantom Fox MS until MA-24 Shining God MS (Takara's final release) or "Bearing Survivor" (Jiraiya MS, Hasbro's final release), as well as any other older models produced during this period.
Easily the most loose fitting, which is not ideal at all for battle. Having a loose BP can cost you your RC prongs, as when hit the BP can be pushed up against the prongs with such force they snap. Just like staying away from BP and RC combos that are too tight in Small type, too loose is equally dangerous.
This isn't set in stone, there are some configurations where a Large type may fit perfectly for one of your RCs, and in those cases you are recommended to use them, just like any other BP type. In general, earlier released RCs with Mold 1 tend to be the most loose with later released Large Type BPs.
Be sure to test out each type on a given RC to see which gives you the best balance of grip and give, not shaky but not hard to twist either.
Diameter of "Large Type"/"Mold 2": 9.1mm-9.2mm
Two Large Mold BP: Magical Ape MS' and Shining God MS'
Shooter Change Core BPs
Because these two RC's have the WD attached beneath a plastic divider, right above the tip, a Bit Protector casing was made specifically to lock it into place without obstructing the RC's tip. To achieve this, they remade the HMS BP with a hole in the center to create the same kind of locking mechanism present in normal HMS anatomy while also allowing a space for the tip to go through.
Shooter Change Core Alpha and Gamma "BP".
Rubber Weight Core's BP
Found in: MA-19 Round Shell MS.
When you intend to use Rubber Weight Core outside of an HMS customization, and instead in the Running Shooter configuration, this piece comes into play. This configuration has zero practical value for Beyblade at all, it simply allows you to shoot this RC vertically outwards, like a wheel. We don't know why they decided this was relevant.
Regardless, to do this, you would have to remove the AR and BP of Round Shell MS, and instead lock into place the specific Running Mode BP that has the same locking mechanism as a regular BP, but a small pole extending out of it, to allow the Shooter to latch onto both sides of the Rubber Weight Core before revving and launching.
Gyro Mode's BP
Found in: MA-19 Magical Ape MS.
This unique BP is found inside of Magical Ape MS' Gyro Mode. It is meant to lock an HMS AR in place, between a shaft (hence the hole) and bearings. Removing the Gyro AR and the locking mechanism at the Mode's free-spinning tip allows for the dissection of the spinning top, where you will find said shaft and bearings, and be able to put in an AR of your choosing. In terms of performance, whichever AR you pick to place inside will have little impact on the viability of this gimmick top, unfortunately.