You don't show the full code, but my guess is, you set and enable the callback after the timer has already elapsed once or many times, so there is a pending elapsed event which triggers immediately when you try to enable the functionality. It may be due to you using debugger and the timer runs even if code is stopped. It may be that you enable the callback after timer has alredy been running due to spending time running any other code before enabling the callback.

And why the HAL seems to wait one extra millisecod; it can be simply explained by looking at the source code of HAL_Delay - surprised that your source of information did not bother to check it and left it as mystery.

First of all, the HAL code adds 1 timer tick, and as they happen at 1ms in your system, asking for a 50ms delay will guarantee you get a delay of 51 ticks of the timer that increases the time variable.

The comment also explains why it adds the 1 timer tick. As when you call the HAL_Delay with some parameter like 50, you likely call it at a random time compared to the system timer ticks. So there can be a time range between 0 and 1 milliseconds before the next tick happens. If you want a 50ms delay, the HAL is guaranteed to give you a 50ms delay, in the range of 50 to 51 milliseconds, by adding one more tick to wait. If it did not do this, then you would be asking why you get time delays between 49 and 50 milliseconds even if you asked for 50 because you thought it will be enough and you get exactly 50ms but no less. It is hard to please everyone and with 1ms timer ticks, you only get 1ms precision or granularity in the times you need to delay.

I had a similar situations and after a bit of digging traced it down to this statement in the ST drivers IRQ Handler, the same that you have included in the question:

if(__HAL_TIM_GET_IT_SOURCE(htim, TIM_IT_UPDATE) !=RESET)

Confusingly named, this line is actually checking whether the Update callback is enabled.

I therefore was able to trigger the callback once I enabled the TIM_IT_UPDATE feature...

This can be done in two ways (links are referencing the G4 HAL drivers):

Using the HAL_TIM_Base_Start_IT() (Not Recommended)

Which enables this feature as a consequence

HAL_TIM_Base_Start_IT(htim)

Note: This is only "not recommended" as you are using the OC mode

Using __HAL_TIM_ENABLE_IT()

Running this in an initializing section (e.g. HAL_TIM_OC_MspInit()) :

__HAL_TIM_ENABLE_IT(htim, TIM_IT_UPDATE);

Notes

I would probably suggest the second option if it works for you, as the ST docstring suggests the former is for setting the timer as a Timer base, which may have unintended consequences for this use case.