First of all, I would start off presenting yourself: Dear XYZ support team I am the web developer in charge of example.com website. By presenting you this way, you are establishing the frame to treat you, hinting that you should be presupposed to be somewhat proficient, so they could choose to answer in a more technical detail.

Feel free to hint, brag, or both! The best CS reps should easily take a hint from clear language and a signature like John Appleseed, JavaScript/Ruby Developer, but any will catch on with a simple line like "I know what I'm doing, so I'd appreciate an extra-technical explanation!"

Hey, I'm a PhD in [field] and do [whatever] professionally. Before calling you, I've narrowed down the problem to [something on their end], so that's what needs to be addressed. If I could speak to an engineer about [specific problem], that'd be great; but if we've gotta walk through the script, let's just knock it out quickly. If they end up requiring the script, then the best way to use your expertise is to run through it quickly. Keep the chit-chat to a minimum and just do the stuff efficiently. If they start describing how to perform some step, you might interrupt them with, "Got it, just a sec.", then let them know once you're ready for the next step.

However, what speaks against just straight up telling them that you're working as [Insert title of your position] and you know what you're talking about?

In one of my internship, I got to befriend a level 2 tech support, so learned a couple thing of how it worked (in that company). Level 1 was out-sourced, and they had a script to go from, regularly updated. From statistics, this took care of 90% of issues. Level 2 was a double handful of tech people, they had basic troubleshooting tools and knowledge and would solve 90% of the remaining issues. Level 3 was the engineering department (where I was), and as a result of level 1 and 2 efficiency less than 1% of issues ever got escalated. The process worked!

OP is referring to letting people know they can speak like proper adults when talking about technical terms, without going through the usual nanny-like discourse that tech support has to provide to non-techies. For instance, it happened to me with Amazon support. The speaker told me exactly where to touch in order to clear the cache of the Android Amazon App Store. Given that I work as an app developer the guy could have just said "please clear the cache". No need to go through "tap here, then here, now you should see this, tap that"...

I have tried different tactics of showing the tech support that I am proficient in their field of work (I work as a web developer). Specifically: using accurate terms and technologies to show my knowledge of them and telling the support that I am the "administrator" of the website in question.

How to let tech support subtly know that I am proficient without showing off?

Unfortunately the tech support people you are speaking to are probably as frustrated as you are at having to go through the basic stuff with you.

Large companies especially deal with the massive volume of tech support calls they receive by employing some staff on lower pay as a "buffer," dealing with simple or "known" issues so that they don't need to employ as many higher paid "second line" support staff.

Very often the first people you get through to on tech support lines are reading from a script.