In D&D, “diamond” and “crystal” are probably not the same thing
As per the description in the Player’s Handbook, an arcane focus is:
a special item ... designed to channel the power of arcane spells.
It costs between 5 and 20 gold pieces, with a crystal type costing 10gp. This indicates that a crystal of suitable size and properties, worked by someone with the appropriate skills into a form that can channel magic, is worth that much. So the materials from which it’s made can’t be that expensive.
So far, so good. But what does “crystal” mean in this context? In a modern real world sense, the term has a scientific meaning: a pure substance whose molecules are arranged in a geometric pattern. That absolutely does include everything from quartz to diamonds and rubies, but also sugar and salt.
In common usage, though, you wouldn’t call a diamond or other gemstone a “crystal”. In fact “a crystal” is more commonly understood to be something less valuable: quartz, amethyst, or even - as per the other modern English dictionary definition - “highly transparent glass with a high refractive index”, as used in chandeliers, fancy glassware and costume jewellery.
The spell doesn’t require “a crystal”, but a diamond worth - on its own - 50gp. That might be a pretty small diamond (though that’s open to interpretation), and it might not even be cut or polished. It’s a material substance required by the spell, though as it is not consumed by the spell, it could be described as channeling the energy of the magic rather than as an ingredient - spells often leave the details of what that looks like up to the player.
It’s up to your DM, but a diamond arcane focus should probably cost more than 50gp
So where am I going with all this? I think it’s reasonable to say yes, you can have a diamond crystal focus usable for this spell - but you’d need to start with a diamond worth at least 50gp and then pay more for someone with the right skills to turn it into an arcane focus. The simplest solution would be to buy one and have it cost 60gp, combining the costs involved, but a DM might also consider this a great opportunity for a side quest - finding a diamond with “the right properties”, or an artisan with the right skills etc.
But equally, given we’re talking about what words mean in D&D and not the real world, it would be reasonable to think that when folks in the Forgotten Realms or other “magical medieval Europe” style settings say “crystal” they mean a lump of quartz or polished glass - certainly, given the cost of a crystal arcane focus, it doesn’t seem they mean a precious gemstone. Even in real life no-one wants a “crystal wedding ring”, even if that’s scientifically accurate. And in various treasure tables and other contexts in the game rules, you find the terms “diamond”, “gemstone” and “crystal” used separately - not interchangeably - indicating they are meant to be different things.
So a DM would also be within their rights to say that for D&D worlds, “crystal” and “diamond” are not the same thing. They would also be justified in allowing this, but with a caveat that such a specialised bit of equipment would cost significantly more than 60gp - after all, working diamond is much more difficult than working glass or quartz.