Try to improve counter picking skills

Counterpicking is one of the most overrated tactics in Dota 2.

Many "Counterpicks" Aren't Counterpicks

• Myth: They have Riki, therefore I should pick Bounty Hunter or Slardar so we have vision of him.
• Reality: A smart player isn't going to auto attack some creeps, then initiate on you so you have time to (a) see him, (b) cast your true sight ult. Reactionary/Premeditated ults like Amplify Damage and Track won't prevent a Riki from initiating onto you, using cloud to silence, and ripping you apart. Only good team play, map awareness, and use of Sentry Wards/Gem will help.

And because I know Dota community is nit-picky:  Amplify Damage/Track/Dust/Etc. can help if you are on the offensive, and ganking a braindead Riki with no map awareness that's farming the lane.

Many "Counterpicks" Can Hurt More Than They Help

A hero like Timbersaw is a pretty good way to deal with many STR carry/semi-carry heroes because of the 15% stat reduction with Whirling Death.

However, Timbersaw needs a solo lane. If you already have a mid and offlane, Timbersaw will be crippled by lack of farm and XP. In the end, he will hurt your team composition rather than helping it.

You Don't Have to Deal With Their Heroes to Win

You win the game by hitting buildings. In many situations, you can just get them responding to your pushes and the carries they chose will have difficulty farming when their map control is reduced to jack shit.

Let's say you pick Pugna & Leshrac + 1 for your safe trilane. Then get some mid and offlane.

You're going up against let's say a Bloodseeker and Lifestealer. Traditionally, all that magic damage looks bad against Lifestealer who has built in Black King Bar. You also have several squishy heroes against a Bloodseeker.

However, with those 2 heroes you can bulldoze their T1 tower in < 5 min. Then proceed to the T2. If there are no TPs, then you'll be at their base in under 10 minutes.

If there are TPs, now your offlaner gets a 1v1 lane against Lifestealer, when your offlane already has a level advantage from being 3v1.

If Bloodseeker attempts to gank, it doesn't matter because you have 3 heroes bunched up. If Pugna rushes a Mekansm (which you normally would in that situation), then any gank attempt will be pretty futile.

When It Makes Sense to Counterpick

Only counterpick when it fits in with your original plan, or you can work your plan around it.

For instance, let's say you were going to pick some sort of semi-carry for mid. The opposing team has Outworld Devourer and Lifestealer. Now it makes sense to pick Razor because he's considered a counter to both heroes.

If your team needs a playmaker, then you might have to go with Puck (who Outworld Devourer is strong against) and pick up strong ganking supports so that Puck can maintain some control in the lane via smoke ganks froom the supports.

Regardless of whether you're drafting in a captain's game or just choosing your own hero in AP, I think it's generally much better to take a high level view rather than trying to counter specific heroes. If your opponent picks someone who's obviously going to be in a certain lane (for example Outworld Devourer mid) then feel free to pick accordingly, but most of the time I'd say just look at the kind of heroes that are showing up in the opposing draft, and try to pick yours with that in mind.

For example if they're drafting heroes with a lot of burst damage, you want to avoid picking anyone too fragile. It's not so much a case of 'they picked Zeus, he's countered by Storm Spirit (to take an example from the site you linked) so much as Zeus and Timbersaw have a lot of burst damage, I probably shouldn't pick Crystal Maiden.

Similarly, think about how they're going to be playing the game

If they're picking heroes who like to fight early, don't choose someone who needs 40 minutes of farm to be useful. That applies to your own team's picks as much as it does to your opponents.

There are some heroes that are worth counter-picking specifically

The best examples that I can think of are Nature's Prophet and Broodmother - these guys will split push you to death if you let them, so you need to make sure that your team has some kind of response to that. the first thing a new drafter should do for his team is figure out what heroes the team can actually play well. 90% of the time, a good hero played well will do better than the perfect hero played poorly.

Once you have a list of heroes that everyone can play (at least 3-4 per player that they are "good" at and another 2-3 that they understand the basics of) make up a list of 4 ideal teams for the core playstyles: ganknig aggression, 4 protect 1, 5 man push, and split push. Maybe also throw in a wombo combo team as well for when you want to rely on winning silly teamfights with stuff like  Reverse >  Ghost Ship >  Split Earth > Macropyre. In each other those 30 hero slots, fill in 2 alternates from your pool of workable heroes. They don't need to synergize perfectly, but you should understand how they will affect the team. Switching lesh for shadow shaman gives you much more consistent lockdown but you lose a lot of early damage and you can't really take 5 minute towers anymore.

Going into a game, know which of the team plans you are going to use

Start your draft by banning out T1 heroes that you think will erally mess with your plan. For example, if you want to split push, you might want to ban out clockwerk. If you are going to rely on Puck mid having a good game, you can ban out  Outworld Devourer. After a few games, you should figure out what heroes give your team comp the most trouble.

In your first round picks, you either want to get your most versatile heros (like alchemist, clockwerk or mirana) or a hero that is 100% core to the team (like nature's prophet in a split push team or visage in a really aggressive team)

If you pick versatile heroes, the enemy has very little information to ban with and doesn't have much idea how your lanes will look. If you take a core hero first, you know it won't get banned, but the enemy team has a better idea of what your game plan is and can draft around that.

Second ban phase is where the draft gets interesting

Both sides know some information about the enemy comp and can choose to either target bans on heroes that make sense in the enemy team or you can keep banning heroes that acn mess with your comp. What heroes you ban depeneds a lot on your skill as a drafter, and you will only get that with experience.

The second ban phase is where your flexibility comes in

You should have a pretty good idea what their game plan is, certainly after their 3rd pick. Here, you can swap in some of the alternate heroes you planned for. Does it look like they have a very squishy team? Lina could be a good alternative to Jakiro. Does it look like a splitpush lineup? Pick up someone who can set up your ganks well like Nyx Assassin. Just remember your overall gameplan. If you think your original team will beat their gameplan, it is up to them to make a correction, not you.

Last ban is tough

There are often 2-3 good heroes for the remaining slot on their team, or even more if you think they are running duo lanes or aggressive trilane. Just get rid of whatever you think will be most annoying and that would fit in their comp.

Last pick should be saved for the hero you think is most counterable

Often this will be your mid or carry hero (Death Prophet is a special case because frankly letting her through the first round ban rarely happens in pubs and is basically an inviation to in an early push game). Many games are won or lost based on which team comes out ahead in the mid lane, but just as many games are won based on how soon the carries participate and how late they peak off. It is up to you to prioritize which you want to use your last pick on.