Louise, welcome back to the wait, and the short answer to your first question is yes: a home test on the official day is a completely normal way for a clinic to run test day, it's how a great many clinics do it, and it isn't a cheaper-clinic thing. My own official test on our second round was a bathroom, a shop test, and a phone call, and our third round, at the same clinic, was a blood test, only because I asked for one out of pure nerves. Same clinic, same standard of care, two ways of getting the same answer.
The reason it works is timing. The clinic sets your date, usually 9 to 14 days after transfer depending on whether you had a day 3 or day 5 embryo, so that if the embryo has implanted the hCG level is comfortably above what a home test can detect, and so that the hCG from your trigger injection has long since cleared. That's the whole reason they were so firm about not testing early: on the official day the stick is reliable, and before it, it isn't. The site's guide to the two-week wait explains why the exact date is chosen the way it is.
What a blood test adds is a number, and Dhruv is right about what the number is for. A beta hCG on its own is a poor predictor because the normal range on test day is enormous; what clinics use it for is the trend, which is why a positive blood test usually comes with a repeat about 48 hours later. In a healthy early pregnancy hCG roughly doubles every two to three days at this stage, so a level that's rising well is reassuring and one that's flat or falling tells them to look more closely. Clinics that use home tests skip that step for a clear positive and go straight to booking the early scan, which is the thing that really answers the question: it's usually around 6 to 7 weeks of pregnancy, so about 4 to 5 weeks after a day 5 transfer. A blood test is what they reach for when the picture is unclear, which brings me to your third question.
A faint line on the official day: you ring and you say "there's a faint line". That's all. A faint line on test day is a positive to them, not a maybe, and what they'll almost certainly do is ask you to keep taking your progesterone exactly as you have been, and either book the scan or, if they want more information, bring you in for the blood test you were hoping for anyway. The one outcome nobody warns you about is a positive that fades over the following days, which the clinic will call a biochemical or chemical pregnancy: an embryo that implanted and then stopped very early. It's a real loss and it deserves to be treated as one, and it's also the case where a blood trend is genuinely useful, so if that happens the clinic will be testing your blood then whether or not they did on Tuesday.
Two small practical things from someone who's stood in that bathroom. Use first morning urine and a test you've bought fresh, not the ones that have been living at your mum's since May, and read it in the time window on the box rather than an hour later. And decide tonight who's awake with you on Tuesday; that one piece of planning did more for me than anything. Anything about your own result or your own protocol is for your clinic, and they will genuinely not mind hearing from you at 8am with a description of a line.