@Fressia123, I suppose I may be naive but I think you are handling this in exactly the right way.
I believe, based on what you have written, that there is an excellent chance that you are at the beginning of a wonderful stage of real progression in your career with your current company. I disagree with the posters who continue to say you are making a mistake by not jumping to this job that you don’t really want.
That other employer has already been deceptive by telling you only towards the end of the process that it’s a nighttime job. Who knows what other surprises they might spring, should you leave your job and move over there? Maybe they expect you on call 24/7 to answer emails or even to go in to the site at a moment’s notice. Maybe the job content will be unachievable without working unlimited “flexible” hours. They could require you to cancel any holiday at any time, with no notice, at your own expense. There may be legal limits on what they can literally demand, but companies can get around that by setting “performance expectations” that cannot otherwise be met, unless you voluntarily do all of these things for them.
Meanwhile, your current employer now knows that your CV and interview profile were so good that you got offered a whole lot more money somewhere else doing a completely different job. They also know that you a genuine commitment to the industry and company, making it more potentially worthwhile for them to invest in you. They know that at least one headhunter has identified you as a promising candidate for daytime roles for which they may be recruiting, and that you have the capability and ambition to take on more responsibility and thus justify a higher salary than what you are getting now. Contrary to a recent previous poster, I just don’t see how all of that that gives your employer more leverage to keep you in your place and to stop your progression!
Of course there is a risk that they will be unwilling or unable to keep their promises, but if that happens, then you will still be free to leave them for another job as and when they fail to follow through on anything that have promised. Whereas if you stay, you have a written list of promises that makes it much harder for them to feign innocence. They are raising your salary effective very soon, and it sounds as though they are making real investments into your career right now, by having you shadow people in other roles, in order to help you get ready for a new role with your current firm. Of course you need to watch closely whether they follow through, but this really does not sound like the set of actions one would expect from a CEO would take who is planning to let you down at the last moment. Instead, what you are hearing and seeing sounds very promising, and will become more so assuming it stays on track.
Best of luck, congratulations, and well done for having earned this opportunity with your existing firm!