> ...they thought I should be using the S word!
The question suggests that you can somehow give yourself a new title which usually isn't true for people who work for someone else. Hopefully the titles that you put on your resume are your actual titles at the companies that you worked for, not something you came up with. Don't worry about calling yourself anything.
At the same time if you are able to get a promotion or apply for a senior role, that's great - go for it. It's nice that people around you are recognising your progress in some way.
Last thing, titles have meaning within any given company but are almost useless when comparing across companies. In some places to get a senior developer title, you just need to ask. Your manager then tells HR to change one cell in a spreadsheet and congrats - you're a senior software engineer. In other places you can become senior if you've been on a team for a year and everyone else has been there for a couple of months ("you have multiples of their experience, of course we can call you a senior engineer"). In those places you just need to be senior relative to the people around you. And in other companies it's a much more rigourous process with very specific criteria. There are people will decades of impressive high-profile engineering work at big companies whose title is just "Senior Software Engineer".