For the prescription shown, it appears to say TFCC injury, right wrist.

Best choice: use a rigid wrist splint with a palmar metal stay, ideally one that also limits forearm rotation if pain is on the little-finger side or there is distal radioulnar joint instability.

Of the two products shown:

Ask for a right-sided “wrist and forearm immobilizer / TFCC brace”. A hand specialist may instead prescribe a Muenster-type brace or temporary long-arm splint when rotation of the forearm is painful or the joint feels unstable. TFCC-related ulnar wrist pain is commonly managed initially with bracing or casting, as noted by the American Society for Surgery of the Hand.

Wear it comfortably tight, not tight enough to cause numbness, tingling, cold fingers, or color change. Avoid twisting movements, heavy gripping, push-ups, and lifting until reviewed by your orthopedist.

Since the pain is mild, choose the Tynor Wrist Brace with Double Lock shown in the second photo.

It supports the wrist while allowing the thumb and fingers to move. Use the right-hand version and wear it during activities that trigger pain, such as lifting, twisting a key, opening jars, gym work, or long phone use.

You likely do not need the thumb-support brace unless pain is mainly at the thumb side/base of thumb.

Practical use:

If pain is on the little-finger side and does not improve in 1-2 weeks, or if there is clicking, weakness, swelling, or pain with rotating the forearm, see the orthopedist again.

That location is the ulnar-dorsal side of the wrist, near the ring and little-finger side. It fits with the area affected in a possible TFCC irritation, but can also arise from the ECU tendon on the back-outside of the wrist. Both are commonly aggravated by twisting, lifting, pushing off from a chair, and forceful gripping.

For mild pain, choose the Tynor Wrist Brace with Double Lock from your second photo, on the right wrist. It is the appropriate one of the two because it supports the wrist without unnecessarily immobilising the thumb.

Use it:

Do not use the thumb brace as the primary support unless there is separate thumb pain.

Arrange orthopaedic or hand-specialist review sooner if you have a painful click/clunk, visible swelling, reduced grip, numbness in the ring/little fingers, pain after a fall, or no clear improvement within 1-2 weeks. Ulnar-sided wrist pain has several possible causes, so a brace can reduce symptoms but does not confirm TFCC as the cause.

You are right. A double-lock wrist brace only supports the wrist, so it may not adequately protect pain that extends onto the back of the hand near the ring and little-finger metacarpals.

For that pain location, ask for a:

It supports:

It is more appropriate than either product in the photos if the painful spot is on the hand itself. A short-arm ulnar gutter splint immobilizes the wrist plus the ulnar half of the hand. It is commonly used for injuries involving the ring/little-finger side of the hand. Hand splint guidance

For mild pain, use it only during aggravating activity or short periods for a few days, and avoid tight, continuous immobilization without an examination. Since this location can also indicate a 4th/5th metacarpal injury or tendon/ligament irritation, get an X-ray or clinical review promptly if there was a punch, fall, direct blow, swelling, marked tenderness over a bone, bruising, altered finger alignment, or pain persists beyond 5-7 days.