Matter: Linkind bulb Power Up State setting is not applied, but works when configured through AiDot

### The problem

I have a Linkind A19 CCT Matter-over-Wi-Fi bulb commissioned to Home Assistant.
Home Assistant exposes a "Power Up State" configuration entity with On, Off, Toggle, and Previous State options. Changing this setting in Home Assistant appears successful in the UI, but the selected startup behavior is not honored after an AC power cycle.
Reproduction:
1. Set Power Up State to "Off" in Home Assistant.
2. Wait at least 1-2 minutes.
3. Remove AC power from the bulb.
4. Restore AC power.
5. The bulb powers ON instead of remaining OFF.
The same occurs with "Previous State". If the bulb is turned OFF through Home Assistant before AC power is removed, it still powers ON when AC power is restored.
I then added the same bulb to AiDot using Matter Multi-Admin, without factory resetting, removing it from Home Assistant, or changing its firmware.
When I configure the equivalent Power Up State through AiDot, the bulb correctly honors the setting after an AC power cycle.
Therefore, with the same bulb and firmware:
- Power Up State configured through Home Assistant -> bulb doesn't honor the state selected
- Power Up State configured through AiDot -> selected startup behavior works correctly.

AI analysis of the diags, show Home Assistant Matter diagnostics also show endpoint 1, On/Off cluster (0x0006), StartUpOnOff (0x4003 / decimal 16387) with a value of 0 (Off), despite the bulb powering ON after AC power is restored.
Device information from Matter diagnostics:
- Manufacturer: Leedarson
- Product: Linkind Smart Light Bulb
- Vendor ID: 5014
- Product ID: 4120
- Hardware version: V1.0.0
- Software version: 1.00.00
This may indicate that the Matter startup configuration written by Home Assistant differs from what AiDot configures, or that AiDot is using an additional/vendor-specific mechanism.

### What version of Home Assistant Core has the issue?

core-2026.9.1

### What was the last working version of Home Assistant Core?

core-unknown

### What type of installation are you running?

Home Assistant OS

### Integration causing the issue

Matter

### Link to integration documentation on our website

https://www.home-assistant.io/integrations/matter/

### Diagnostics information

[matter-01JGAQ1SHAYSCVXJTNSGP9PGCS-Linkind Smart Light Bulb-72e2bceb595c0cfc564a859dbf5de7ad.json](https://github.com/user-attachments/files/32028045/matter-01JGAQ1SHAYSCVXJTNSGP9PGCS-Linkind.Smart.Light.Bulb-72e2bceb595c0cfc564a859dbf5de7ad.json)

### Example YAML snippet

```yaml

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### Anything in the logs that might be useful for us?

```txt

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### Additional information

_No response_

Start with the Matter integration entry point and the attached diagnostics, focusing on endpoint 1's On/Off cluster and the StartUpOnOff attribute (0x4003). Compare the value written by Home Assistant with the AiDot behavior across an AC power cycle; done means identifying the differing handling and verifying that the selected startup state is honored.

Written by the indexing model from the issue text.