rough shifting problem with Tiagra

Having a work-stand makes this sort of adjustment soooooo much easier.
They are a worthwhile investment - although if you don't want to spend out now, perhaps someone local to you has one they could lend you?

From the pictures - that looks like a long reach cage? What cassette are you running?

When you are changing 'up' to the small cogs, and they don't engage right - have you done a visual check to make sure the rear mech cage is directly below the right cog?

If the mech cage is not aligned quite right, you may need to use the barrel adjuster on the rear mech to adjust the position so it changes cleanly.

Form my experience with Tiagra, its a good groupset, and works well - but it needs to be adjusted 'just so'. When its set right, it works well, but if its only slightly out, its horrible. I always find that bad changes into the smaller cogs (as you describe) are usually the first warning that rear mech is slightly out of alignment.

To my eye, the chain length looks fine. If you have a quick link, you might want to check using the big chainring and big rear cog + 2 links measurement.

Daft question but I assume its a 10 speed chain?

Yep. It's a good groupset apart from the Chainset looking a bit rubbish. Ran it for 2 years on the winter bike without fault. Hope you find the gremlin in the works,

Chain does look a bit long but it depends if that video is when the chain is on the large or small front ring. If large chain is definitely too long.

Length of rear cable and radius of the loop is very short so Mike the Bike could be right. Does your version of Tiagra have gear cables that run under the bar tape or the old fashioned side-exit type from the shifters? The latter are far better for shifting by maintaining less tension in the cable compared to more modern, albeit better looking versions. My Tiagra 4600 shifters shift better than the 6700 Ultegra shifters which have a more convoluted cable run.

Check your "B" screw, looking at the angle of the cage, you do not have enough tension on your chain. The cage should be almost vertical, not layer over like it is. Or your chain is too long, but check the "B" screw first.

On a personal note, you are seriously in violation of Rule 49.

Bent hanger?

This may be a long shot but it's cheap to try ....

Several years of Tiagra have convinced me it's brilliant value for money but the consistency of the changes depends on the alignment of the gear cable. I think the last section of your cable, from the chain stay to the derailleur, is too short. A more gentle curve might very well transform your shifting, at least it has worked for me in the past.

Best of luck.